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Library of Congress Cataloging-in-Publication Data
Kleinberg, Jon.
Algorithm design / Jon Kleinberg, Éva Tardos.—1st ed.
p. cm.
Includes bibliographical references and index.
ISBN 0-321-29535-8 (alk. paper)
1. Computer algorithms. 2. Data structures (Computer science) I. Tardos, Éva.
II. Title.
QA76.9.A43K54 2005
Copyright © 2006 by Pearson Education, Inc.
For information on obtaining permission for use of material in this work, please
submit a written request to Pearson Education, Inc., Rights and Contract Department,
75 Arlington Street, Suite 300, Boston, MA 02116 or fax your request to (617) 848-7047.
All rights reserved. No part of this publication may be reproduced, stored in a
retrieval system, or transmitted, in any form or by any means, electronic, mechanical,
photocopying, recording, or any toher media embodiments now known or hereafter to
become known, without the prior written permission of the publisher. Printed in the
United States of America.
ISBN 0-321-29535-8
1 2 3 4 5 6 7 8 9 10-CRW-08 07 06 05
About the Authors
Jon Kleinberg is a professor of Computer Science at
Cornell University. He received his Ph.D. from M.I.T.
in 1996. He is the recipient of an NSF Career Award,
an ONR Young Investigator Award, an IBM Outstanding Innovation Award, the National Academy of Sciences Award for Initiatives in Research, research fellowships from the Packard and Sloan Foundations,
and teaching awards from the Cornell Engineering
College and Computer Science Department.
Kleinberg’s research is centered around algorithms, particularly those concerned with the structure of networks and information, and with applications
to information science, optimization, data mining, and computational biology. His work on network analysis using hubs and authorities helped form the
foundation for the current generation of Internet search engines.
Éva Tardos is a professor of Computer Science at Cornell University. She received her Ph.D. from Eötvös
University in Budapest, Hungary in 1984. She is a
member of the American Academy of Arts and Sciences, and an ACM Fellow; she is the recipient of an
NSF Presidential Young Investigator Award, the Fulkerson Prize, research fellowships from the Guggenheim, Packard, and Sloan Foundations, and teaching awards from the Cornell Engineering College and
Computer Science Department.
Tardos’s research interests are focused on the design and analysis of
algorithms for problems on graphs or networks. She is most known for her
work on network-flow algorithms and approximation algorithms for network
problems. Her recent work focuses on algorithmic game theory, an emerging
area concerned with designing systems and algorithms for selfish users.
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About the Authors
Introduction: Some Representative Problems
A First Problem: Stable Matching 1
Five Representative Problems 12
Solved Exercises 19
Exercises 22
Notes and Further Reading 28
Basics of Algorithm Analysis
Computational Tractability 29
Asymptotic Order of Growth 35
Implementing the Stable Matching Algorithm Using Lists and
Arrays 42
A Survey of Common Running Times 47
A More Complex Data Structure: Priority Queues 57
Solved Exercises 65
Exercises 67
Notes and Further Reading 70
Basic Definitions and Applications 73
Graph Connectivity and Graph Traversal 78
Implementing Graph Traversal Using Queues and Stacks 87
Testing Bipartiteness: An Application of Breadth-First Search 94
Connectivity in Directed Graphs 97
Directed Acyclic Graphs and Topological Ordering
Solved Exercises 104
Exercises 107
Notes and Further Reading 112
Greedy Algorithms
Interval Scheduling: The Greedy Algorithm Stays Ahead 116
Scheduling to Minimize Lateness: An Exchange Argument 125
Optimal Caching: A More Complex Exchange Argument 131
Shortest Paths in a Graph 137
The Minimum Spanning Tree Problem 142
Implementing Kruskal’s Algorithm: The Union-Find Data
Structure 151
Clustering 157
Huffman Codes and Data Compression 161
∗ 4.9
Minimum-Cost Arborescences: A Multi-Phase Greedy
Algorithm 177
Solved Exercises 183
Exercises 188
Notes and Further Reading 205
Divide and Conquer
A First Recurrence: The Mergesort Algorithm
Further Recurrence Relations 214
Counting Inversions 221
Finding the Closest Pair of Points 225
Integer Multiplication 231
Convolutions and the Fast Fourier Transform
Solved Exercises 242
Exercises 246
Notes and Further Reading 249
Dynamic Programming
Weighted Interval Scheduling: A Recursive Procedure 252
Principles of Dynamic Programming: Memoization or Iteration
over Subproblems 258
Segmented Least Squares: Multi-way Choices 261
The star indicates an optional section. (See the Preface for more information about the relationships
among the chapters and sections.)
∗ 6.10
Subset Sums and Knapsacks: Adding a Variable 266
RNA Secondary Structure: Dynamic Programming over
Intervals 272
Sequence Alignment 278
Sequence Alignment in Linear Space via Divide and
Conquer 284
Shortest Paths in a Graph 290
Shortest Paths and Distance Vector Protocols 297
Negative Cycles in a Graph 301
Solved Exercises 307
Exercises 312
Notes and Further Reading 335
Network Flow
The Maximum-Flow Problem and the Ford-Fulkerson
Algorithm 338
Maximum Flows and Minimum Cuts in a Network 346
Choosing Good Augmenting Paths 352
∗ 7.4
The Preflow-Push Maximum-Flow Algorithm 357
A First Application: The Bipartite Matching Problem 367
Disjoint Paths in Directed and Undirected Graphs 373
Extensions to the Maximum-Flow Problem 378
Survey Design 384
Airline Scheduling 387
7.10 Image Segmentation 391
7.11 Project Selection 396
7.12 Baseball Elimination 400
∗ 7.13
A Further Direction: Adding Costs to the Matching Problem
Solved Exercises 411
Exercises 415
Notes and Further Reading 448
NP and Computational Intractability
Polynomial-Time Reductions 452
Reductions via “Gadgets”: The Satisfiability Problem
Efficient Certification and the Definition of NP 463
NP-Complete Problems 466
Sequencing Problems 473
Partitioning Problems 481
Graph Coloring 485
Numerical Problems 490
Co-NP and the Asymmetry of NP 495
A Partial Taxonomy of Hard Problems 497
Solved Exercises 500
Exercises 505
Notes and Further Reading 529
PSPACE: A Class of Problems beyond NP
Some Hard Problems in PSPACE 533
Solving Quantified Problems and Games in Polynomial
Space 536
Solving the Planning Problem in Polynomial Space 538
Proving Problems PSPACE-Complete 543
Solved Exercises 547
Exercises 550
Notes and Further Reading 551
10 Extending the Limits of Tractability
∗ 10.4
∗ 10.5
∗ 11.7
Finding Small Vertex Covers 554
Solving NP-Hard Problems on Trees 558
Coloring a Set of Circular Arcs 563
Tree Decompositions of Graphs 572
Constructing a Tree Decomposition 584
Solved Exercises 591
Exercises 594
Notes and Further Reading 598
11 Approximation Algorithms
Greedy Algorithms and Bounds on the Optimum: A Load
Balancing Problem 600
The Center Selection Problem 606
Set Cover: A General Greedy Heuristic 612
The Pricing Method: Vertex Cover 618
Maximization via the Pricing Method: The Disjoint Paths
Problem 624
Linear Programming and Rounding: An Application to Vertex
Cover 630
Load Balancing Revisited: A More Advanced LP Application 637
Arbitrarily Good Approximations: The Knapsack Problem
Solved Exercises 649
Exercises 651
Notes and Further Reading 659
12 Local Search
∗ 12.6
The Landscape of an Optimization Problem 662
The Metropolis Algorithm and Simulated Annealing 666
An Application of Local Search to Hopfield Neural Networks
Maximum-Cut Approximation via Local Search 676
Choosing a Neighbor Relation 679
Classification via Local Search 681
Best-Response Dynamics and Nash Equilibria 690
Solved Exercises 700
Exercises 702
Notes and Further Reading 705
13 Randomized Algorithms
A First Application: Contention Resolution 708
Finding the Global Minimum Cut 714
Random Variables and Their Expectations 719
A Randomized Approximation Algorithm for MAX 3-SAT 724
Randomized Divide and Conquer: Median-Finding and
Quicksort 727
13.6 Hashing: A Randomized Implementation of Dictionaries 734
13.7 Finding the Closest Pair of Points: A Randomized Approach 741
13.8 Randomized Caching 750
13.9 Chernoff Bounds 758
13.10 Load Balancing 760
13.11 Packet Routing 762
13.12 Background: Some Basic Probability Definitions 769
Solved Exercises 776
Exercises 782
Notes and Further Reading 793
Epilogue: Algorithms That Run Forever
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Algorithmic ideas are pervasive, and their reach is apparent in examples both
within computer science and beyond. Some of the major shifts in Internet
routing standards can be viewed as debates over the deficiencies of one
shortest-path algorithm and the relative advantages of another. The basic
notions used by biologists to express similarities among genes and genomes
have algorithmic definitions. The concerns voiced by economists over the
feasibility of combinatorial auctions in practice are rooted partly in the fact that
these auctions contain computationally intractable search problems as special
cases. And algorithmic notions aren’t just restricted to well-known and longstanding problems; one sees the reflections of these ideas on a regular basis,
in novel issues arising across a wide range of areas. The scientist from Yahoo!
who told us over lunch one day about their system for serving ads to users was
describing a set of issues that, deep down, could be modeled as a network flow
problem. So was the former student, now a management consultant working
on staffing protocols for large hospitals, whom we happened to meet on a trip
to New York City.
The point is not simply that algorithms have many applications. The
deeper issue is that the subject of algorithms is a powerful lens through which
to view the field of computer science in general. Algorithmic problems form
the heart of computer science, but they rarely arrive as cleanly packaged,
mathematically precise questions. Rather, they tend to come bundled together
with lots of messy, application-specific detail, some of it essential, some of it
extraneous. As a result, the algorithmic enterprise consists of two fundamental
components: the task of getting to the mathematically clean core of a problem,
and then the task of identifying the appropriate algorithm design techniques,
based on the structure of the problem. These two components interact: the
more comfortable one is with the full array of possible design techniques,
the more one starts to recognize the clean formulations that lie within messy
problems out in the world. At their most effective, then, algorithmic ideas do
not just provide solutions to well-posed problems; they form the language that
lets you cleanly express the underlying questions.
The goal of our book is to convey this approach to algorithms, as a design
process that begins with problems arising across the full range of computing
applications, builds on an understanding of algorithm design techniques, and
results in the development of efficient solutions to these problems. We seek
to explore the role of algorithmic ideas in computer science generally, and
relate these ideas to the range of precisely formulated problems for which we
can design and analyze algorithms. In other words, what are the underlying
issues that motivate these problems, and how did we choose these particular
ways of formulating them? How did we recognize which design principles were
appropriate in different situations?
In keeping with this, our goal is to offer advice on how to identify clean
algorithmic problem formulations in complex issues from different areas of
computing and, from this, how to design efficient algorithms for the resulting
problems. Sophisticated algorithms are often best understood by reconstructing the sequence of ideas—including false starts and dead ends—that led from
simpler initial approaches to the eventual solution. The result is a style of exposition that does not take the most direct route from problem statement to
algorithm, but we feel it better reflects the way that we and our colleagues
genuinely think about these questions.
The book is intended for students who have completed a programmingbased two-semester introductory computer science sequence (the standard
“CS1/CS2” courses) in which they have written programs that implement
basic algorithms, manipulate discrete structures such as trees and graphs, and
apply basic data structures such as arrays, lists, queues, and stacks. Since
the interface between CS1/CS2 and a first algorithms course is not entirely
standard, we begin the book with self-contained coverage of topics that at
some institutions are familiar to students from CS1/CS2, but which at other
institutions are included in the syllabi of the first algorithms course. This
material can thus be treated either as a review or as new material; by including
it, we hope the book can be used in a broader array of courses, and with more
flexibility in the prerequisite knowledge that is assumed.
In keeping with the approach outlined above, we develop the basic algorithm design techniques by drawing on problems from across many areas of
computer science and related fields. To mention a few representative examples
here, we include fairly detailed discussions of applications from systems and
networks (caching, switching, interdomain routing on the Internet), artificial
intelligence (planning, game playing, Hopfield networks), computer vision
(image segmentation), data mining (change-point detection, clustering), operations research (airline scheduling), and computational biology (sequence
alignment, RNA secondary structure).
The notion of computational intractability, and NP-completeness in particular, plays a large role in the book. This is consistent with how we think
about the overall process of algorithm design. Some of the time, an interesting problem arising in an application area will be amenable to an efficient
solution, and some of the time it will be provably NP-complete; in order to
fully address a new algorithmic problem, one should be able to explore both
of these options with equal familiarity. Since so many natural problems in
computer science are NP-complete, the development of methods to deal with
intractable problems has become a crucial issue in the study of algorithms,
and our book heavily reflects this theme. The discovery that a problem is NPcomplete should not be taken as the end of the story, but as an invitation to
begin looking for approximation algorithms, heuristic local search techniques,
or tractable special cases. We include extensive coverage of each of these three
Problems and Solved Exercises
An important feature of the book is the collection of problems. Across all
chapters, the book includes over 200 problems, almost all of them developed
and class-tested in homework or exams as part of our teaching of the course
at Cornell. We view the problems as a crucial component of the book, and
they are structured in keeping with our overall approach to the material. Most
of them consist of extended verbal descriptions of a problem arising in an
application area in computer science or elsewhere out in the world, and part of
the problem is to practice what we discuss in the text: setting up the necessary
notation and formalization, designing an algorithm, and then analyzing it and
proving it correct. (We view a complete answer to one of these problems as
consisting of all these components: a fully explained algorithm, an analysis of
the running time, and a proof of correctness.) The ideas for these problems
come in large part from discussions we have had over the years with people
working in different areas, and in some cases they serve the dual purpose of
recording an interesting (though manageable) application of algorithms that
we haven’t seen written down anywhere else.
To help with the process of working on these problems, we include in
each chapter a section entitled “Solved Exercises,” where we take one or more
problems and describe how to go about formulating a solution. The discussion
devoted to each solved exercise is therefore significantly longer than what
would be needed simply to write a complete, correct solution (in other words,
significantly longer than what it would take to receive full credit if these were
being assigned as homework problems). Rather, as with the rest of the text,
the discussions in these sections should be viewed as trying to give a sense
of the larger process by which one might think about problems of this type,
culminating in the specification of a precise solution.
It is worth mentioning two points concerning the use of these problems
as homework in a course. First, the problems are sequenced roughly in order
of increasing difficulty, but this is only an approximate guide and we advise
against placing too much weight on it: since the bulk of the problems were
designed as homework for our undergraduate class, large subsets of the
problems in each chapter are really closely comparable in terms of difficulty.
Second, aside from the lowest-numbered ones, the problems are designed to
involve some investment of time, both to relate the problem description to the
algorithmic techniques in the chapter, and then to actually design the necessary
algorithm. In our undergraduate class, we have tended to assign roughly three
of these problems per week.
Pedagogical Features and Supplements
In addition to the problems and solved exercises, the book has a number of
further pedagogical features, as well as additional supplements to facilitate its
use for teaching.
As noted earlier, a large number of the sections in the book are devoted
to the formulation of an algorithmic problem—including its background and
underlying motivation—and the design and analysis of an algorithm for this
problem. To reflect this style, these sections are consistently structured around
a sequence of subsections: “The Problem,” where the problem is described
and a precise formulation is worked out; “Designing the Algorithm,” where
the appropriate design technique is employed to develop an algorithm; and
“Analyzing the Algorithm,” which proves properties of the algorithm and
analyzes its efficiency. These subsections are highlighted in the text with an
icon depicting a feather. In cases where extensions to the problem or further
analysis of the algorithm is pursued, there are additional subsections devoted
to these issues. The goal of this structure is to offer a relatively uniform style
of presentation that moves from the initial discussion of a problem arising in a
computing application through to the detailed analysis of a method to solve it.
A number of supplements are available in support of the book itself. An
instructor’s manual works through all the problems, providing full solutions to
each. A set of lecture slides, developed by Kevin Wayne of Princeton University,
is also available; these slides follow the order of the book’s sections and can
thus be used as the foundation for lectures in a course based on the book. These
files are available at www.aw.com. For instructions on obtaining a professor
login and password, search the site for either “Kleinberg” or “Tardos” or
contact your local Addison-Wesley representative.
Finally, we would appreciate receiving feedback on the book. In particular,
as in any book of this length, there are undoubtedly errors that have remained
in the final version. Comments and reports of errors can be sent to us by e-mail,
at the address algbook@cs.cornell.edu; please include the word “feedback”
in the subject line of the message.
Chapter-by-Chapter Synopsis
Chapter 1 starts by introducing some representative algorithmic problems. We
begin immediately with the Stable Matching Problem, since we feel it sets
up the basic issues in algorithm design more concretely and more elegantly
than any abstract discussion could: stable matching is motivated by a natural
though complex real-world issue, from which one can abstract an interesting
problem statement and a surprisingly effective algorithm to solve this problem.
The remainder of Chapter 1 discusses a list of five “representative problems”
that foreshadow topics from the remainder of the course. These five problems
are interrelated in the sense that they are all variations and/or special cases
of the Independent Set Problem; but one is solvable by a greedy algorithm,
one by dynamic programming, one by network flow, one (the Independent
Set Problem itself) is NP-complete, and one is PSPACE-complete. The fact that
closely related problems can vary greatly in complexity is an important theme
of the book, and these five problems serve as milestones that reappear as the
book progresses.
Chapters 2 and 3 cover the interface to the CS1/CS2 course sequence
mentioned earlier. Chapter 2 introduces the key mathematical definitions and
notations used for analyzing algorithms, as well as the motivating principles
behind them. It begins with an informal overview of what it means for a problem to be computationally tractable, together with the concept of polynomial
time as a formal notion of efficiency. It then discusses growth rates of functions and asymptotic analysis more formally, and offers a guide to commonly
occurring functions in algorithm analysis, together with standard applications
in which they arise. Chapter 3 covers the basic definitions and algorithmic
primitives needed for working with graphs, which are central to so many of
the problems in the book. A number of basic graph algorithms are often implemented by students late in the CS1/CS2 course sequence, but it is valuable
to present the material here in a broader algorithm design context. In particular, we discuss basic graph definitions, graph traversal techniques such
as breadth-first search and depth-first search, and directed graph concepts
including strong connectivity and topological ordering.
Chapters 2 and 3 also present many of the basic data structures that will
be used for implementing algorithms throughout the book; more advanced
data structures are presented in subsequent chapters. Our approach to data
structures is to introduce them as they are needed for the implementation of
the algorithms being developed in the book. Thus, although many of the data
structures covered here will be familiar to students from the CS1/CS2 sequence,
our focus is on these data structures in the broader context of algorithm design
and analysis.
Chapters 4 through 7 cover four major algorithm design techniques: greedy
algorithms, divide and conquer, dynamic programming, and network flow.
With greedy algorithms, the challenge is to recognize when they work and
when they don’t; our coverage of this topic is centered around a way of classifying the kinds of arguments used to prove greedy algorithms correct. This
chapter concludes with some of the main applications of greedy algorithms,
for shortest paths, undirected and directed spanning trees, clustering, and
compression. For divide and conquer, we begin with a discussion of strategies
for solving recurrence relations as bounds on running times; we then show
how familiarity with these recurrences can guide the design of algorithms that
improve over straightforward approaches to a number of basic problems, including the comparison of rankings, the computation of closest pairs of points
in the plane, and the Fast Fourier Transform. Next we develop dynamic programming by starting with the recursive intuition behind it, and subsequently
building up more and more expressive recurrence formulations through applications in which they naturally arise. This chapter concludes with extended
discussions of the dynamic programming approach to two fundamental problems: sequence alignment, with applications in computational biology; and
shortest paths in graphs, with connections to Internet routing protocols. Finally, we cover algorithms for network flow problems, devoting much of our
focus in this chapter to discussing a large array of different flow applications.
To the extent that network flow is covered in algorithms courses, students are
often left without an appreciation for the wide range of problems to which it
can be applied; we try to do justice to its versatility by presenting applications
to load balancing, scheduling, image segmentation, and a number of other
Chapters 8 and 9 cover computational intractability. We devote most of
our attention to NP-completeness, organizing the basic NP-complete problems
thematically to help students recognize candidates for reductions when they
encounter new problems. We build up to some fairly complex proofs of NPcompleteness, with guidance on how one goes about constructing a difficult
reduction. We also consider types of computational hardness beyond NPcompleteness, particularly through the topic of PSPACE-completeness. We
find this is a valuable way to emphasize that intractability doesn’t end at
NP-completeness, and PSPACE-completeness also forms the underpinning for
some central notions from artificial intelligence—planning and game playing—
that would otherwise not find a place in the algorithmic landscape we are
Chapters 10 through 12 cover three major techniques for dealing with computationally intractable problems: identification of structured special cases,
approximation algorithms, and local search heuristics. Our chapter on tractable
special cases emphasizes that instances of NP-complete problems arising in
practice may not be nearly as hard as worst-case instances, because they often
contain some structure that can be exploited in the design of an efficient algorithm. We illustrate how NP-complete problems are often efficiently solvable
when restricted to tree-structured inputs, and we conclude with an extended
discussion of tree decompositions of graphs. While this topic is more suitable for a graduate course than for an undergraduate one, it is a technique
with considerable practical utility for which it is hard to find an existing
accessible reference for students. Our chapter on approximation algorithms
discusses both the process of designing effective algorithms and the task of
understanding the optimal solution well enough to obtain good bounds on it.
As design techniques for approximation algorithms, we focus on greedy algorithms, linear programming, and a third method we refer to as “pricing,” which
incorporates ideas from each of the first two. Finally, we discuss local search
heuristics, including the Metropolis algorithm and simulated annealing. This
topic is often missing from undergraduate algorithms courses, because very
little is known in the way of provable guarantees for these algorithms; however, given their widespread use in practice, we feel it is valuable for students
to know something about them, and we also include some cases in which
guarantees can be proved.
Chapter 13 covers the use of randomization in the design of algorithms.
This is a topic on which several nice graduate-level books have been written.
Our goal here is to provide a more compact introduction to some of the
ways in which students can apply randomized techniques using the kind of
background in probability one typically gains from an undergraduate discrete
math course.
Use of the Book
The book is primarily designed for use in a first undergraduate course on
algorithms, but it can also be used as the basis for an introductory graduate
When we use the book at the undergraduate level, we spend roughly
one lecture per numbered section; in cases where there is more than one
lecture’s worth of material in a section (for example, when a section provides
further applications as additional examples), we treat this extra material as a
supplement that students can read about outside of lecture. We skip the starred
sections; while these sections contain important topics, they are less central
to the development of the subject, and in some cases they are harder as well.
We also tend to skip one or two other sections per chapter in the first half of
the book (for example, we tend to skip Sections 4.3, 4.7–4.8, 5.5–5.6, 6.5, 7.6,
and 7.11). We cover roughly half of each of Chapters 11–13.
This last point is worth emphasizing: rather than viewing the later chapters
as “advanced,” and hence off-limits to undergraduate algorithms courses, we
have designed them with the goal that the first few sections of each should
be accessible to an undergraduate audience. Our own undergraduate course
involves material from all these chapters, as we feel that all of these topics
have an important place at the undergraduate level.
Finally, we treat Chapters 2 and 3 primarily as a review of material from
earlier courses; but, as discussed above, the use of these two chapters depends
heavily on the relationship of each specific course to its prerequisites.
The resulting syllabus looks roughly as follows: Chapter 1; Chapters 4–8
(excluding 4.3, 4.7–4.9, 5.5–5.6, 6.5, 6.10, 7.4, 7.6, 7.11, and 7.13); Chapter 9
(briefly); Chapter 10, Sections.10.1 and 10.2; Chapter 11, Sections 11.1, 11.2,
11.6, and 11.8; Chapter 12, Sections 12.1–12.3; and Chapter 13, Sections 13.1–
The book also naturally supports an introductory graduate course on
algorithms. Our view of such a course is that it should introduce students
destined for research in all different areas to the important current themes in
algorithm design. Here we find the emphasis on formulating problems to be
useful as well, since students will soon be trying to define their own research
problems in many different subfields. For this type of course, we cover the
later topics in Chapters 4 and 6 (Sections 4.5–4.9 and 6.5–6.10), cover all of
Chapter 7 (moving more rapidly through the early sections), quickly cover NPcompleteness in Chapter 8 (since many beginning graduate students will have
seen this topic as undergraduates), and then spend the remainder of the time
on Chapters 10–13. Although our focus in an introductory graduate course is
on the more advanced sections, we find it useful for the students to have the
full book to consult for reviewing or filling in background knowledge, given
the range of different undergraduate backgrounds among the students in such
a course.
Finally, the book can be used to support self-study by graduate students,
researchers, or computer professionals who want to get a sense for how they
might be able to use particular algorithm design techniques in the context of
their own work. A number of graduate students and colleagues have used
portions of the book in this way.
This book grew out of the sequence of algorithms courses that we have taught
at Cornell. These courses have grown, as the field has grown, over a number of
years, and they reflect the influence of the Cornell faculty who helped to shape
them during this time, including Juris Hartmanis, Monika Henzinger, John
Hopcroft, Dexter Kozen, Ronitt Rubinfeld, and Sam Toueg. More generally, we
would like to thank all our colleagues at Cornell for countless discussions both
on the material here and on broader issues about the nature of the field.
The course staffs we’ve had in teaching the subject have been tremendously helpful in the formulation of this material. We thank our undergraduate and graduate teaching assistants, Siddharth Alexander, Rie Ando, Elliot
Anshelevich, Lars Backstrom, Steve Baker, Ralph Benzinger, John Bicket,
Doug Burdick, Mike Connor, Vladimir Dizhoor, Shaddin Doghmi, Alexander Druyan, Bowei Du, Sasha Evfimievski, Ariful Gani, Vadim Grinshpun,
Ara Hayrapetyan, Chris Jeuell, Igor Kats, Omar Khan, Mikhail Kobyakov,
Alexei Kopylov, Brian Kulis, Amit Kumar, Yeongwee Lee, Henry Lin, Ashwin Machanavajjhala, Ayan Mandal, Bill McCloskey, Leonid Meyerguz, Evan
Moran, Niranjan Nagarajan, Tina Nolte, Travis Ortogero, Martin Pál, Jon
Peress, Matt Piotrowski, Joe Polastre, Mike Priscott, Xin Qi, Venu Ramasubramanian, Aditya Rao, David Richardson, Brian Sabino, Rachit Siamwalla, Sebastian Silgardo, Alex Slivkins, Chaitanya Swamy, Perry Tam, Nadya Travinin,
Sergei Vassilvitskii, Matthew Wachs, Tom Wexler, Shan-Leung Maverick Woo,
Justin Yang, and Misha Zatsman. Many of them have provided valuable insights, suggestions, and comments on the text. We also thank all the students
in these classes who have provided comments and feedback on early drafts of
the book over the years.
For the past several years, the development of the book has benefited
greatly from the feedback and advice of colleagues who have used prepublication drafts for teaching. Anna Karlin fearlessly adopted a draft as her course
textbook at the University of Washington when it was still in an early stage of
development; she was followed by a number of people who have used it either
as a course textbook or as a resource for teaching: Paul Beame, Allan Borodin,
Devdatt Dubhashi, David Kempe, Gene Kleinberg, Dexter Kozen, Amit Kumar,
Mike Molloy, Yuval Rabani, Tim Roughgarden, Alexa Sharp, Shanghua Teng,
Aravind Srinivasan, Dieter van Melkebeek, Kevin Wayne, Tom Wexler, and
Sue Whitesides. We deeply appreciate their input and advice, which has informed many of our revisions to the content. We would like to additionally
thank Kevin Wayne for producing supplementary material associated with the
book, which promises to greatly extend its utility to future instructors.
In a number of other cases, our approach to particular topics in the book
reflects the infuence of specific colleagues. Many of these contributions have
undoubtedly escaped our notice, but we especially thank Yuri Boykov, Ron
Elber, Dan Huttenlocher, Bobby Kleinberg, Evie Kleinberg, Lillian Lee, David
McAllester, Mark Newman, Prabhakar Raghavan, Bart Selman, David Shmoys,
Steve Strogatz, Olga Veksler, Duncan Watts, and Ramin Zabih.
It has been a pleasure working with Addison Wesley over the past year.
First and foremost, we thank Matt Goldstein for all his advice and guidance in
this process, and for helping us to synthesize a vast amount of review material
into a concrete plan that improved the book. Our early conversations about
the book with Susan Hartman were extremely valuable as well. We thank Matt
and Susan, together with Michelle Brown, Marilyn Lloyd, Patty Mahtani, and
Maite Suarez-Rivas at Addison Wesley, and Paul Anagnostopoulos and Jacqui
Scarlott at Windfall Software, for all their work on the editing, production, and
management of the project. We further thank Paul and Jacqui for their expert
composition of the book. We thank Joyce Wells for the cover design, Nancy
Murphy of Dartmouth Publishing for her work on the figures, Ted Laux for
the indexing, and Carol Leyba and Jennifer McClain for the copyediting and
We thank Anselm Blumer (Tufts University), Richard Chang (University of
Maryland, Baltimore County), Kevin Compton (University of Michigan), Diane
Cook (University of Texas, Arlington), Sariel Har-Peled (University of Illinois,
Urbana-Champaign), Sanjeev Khanna (University of Pennsylvania), Philip
Klein (Brown University), David Matthias (Ohio State University), Adam Meyerson (UCLA), Michael Mitzenmacher (Harvard University), Stephan Olariu
(Old Dominion University), Mohan Paturi (UC San Diego), Edgar Ramos (University of Illinois, Urbana-Champaign), Sanjay Ranka (University of Florida,
Gainesville), Leon Reznik (Rochester Institute of Technology), Subhash Suri
(UC Santa Barbara), Dieter van Melkebeek (University of Wisconsin, Madison), and Bulent Yener (Rensselaer Polytechnic Institute) who generously
contributed their time to provide detailed and thoughtful reviews of the manuscript; their comments led to numerous improvements, both large and small,
in the final version of the text.
Finally, we thank our families—Lillian and Alice, and David, Rebecca, and
Amy. We appreciate their support, patience, and many other contributions
more than we can express in any acknowledgments here.
This book was begun amid the irrational exuberance of the late nineties,
when the arc of computing technology seemed, to many of us, briefly to pass
through a place traditionally occupied by celebrities and other inhabitants of
the pop-cultural firmament. (It was probably just in our imaginations.) Now,
several years after the hype and stock prices have come back to earth, one can
appreciate that in some ways computer science was forever changed by this
period, and in other ways it has remained the same: the driving excitement
that has characterized the field since its early days is as strong and enticing as
ever, the public’s fascination with information technology is still vibrant, and
the reach of computing continues to extend into new disciplines. And so to
all students of the subject, drawn to it for so many different reasons, we hope
you find this book an enjoyable and useful guide wherever your computational
pursuits may take you.
Jon Kleinberg
Éva Tardos
Ithaca, 2005
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Chapter 1
Introduction: Some
Representative Problems
1.1 A First Problem: Stable Matching
As an opening topic, we look at an algorithmic problem that nicely illustrates
many of the themes we will be emphasizing. It is motivated by some very
natural and practical concerns, and from these we formulate a clean and
simple statement of a problem. The algorithm to solve the problem is very
clean as well, and most of our work will be spent in proving that it is correct
and giving an acceptable bound on the amount of time it takes to terminate
with an answer. The problem itself—the Stable Matching Problem—has several
The Problem
The Stable Matching Problem originated, in part, in 1962, when David Gale
and Lloyd Shapley, two mathematical economists, asked the question: Could
one design a college admissions process, or a job recruiting process, that was
self-enforcing? What did they mean by this?
To set up the question, let’s first think informally about the kind of situation
that might arise as a group of friends, all juniors in college majoring in
computer science, begin applying to companies for summer internships. The
crux of the application process is the interplay between two different types
of parties: companies (the employers) and students (the applicants). Each
applicant has a preference ordering on companies, and each company—once
the applications come in—forms a preference ordering on its applicants. Based
on these preferences, companies extend offers to some of their applicants,
applicants choose which of their offers to accept, and people begin heading
off to their summer internships.
Chapter 1 Introduction: Some Representative Problems
Gale and Shapley considered the sorts of things that could start going
wrong with this process, in the absence of any mechanism to enforce the status
quo. Suppose, for example, that your friend Raj has just accepted a summer job
at the large telecommunications company CluNet. A few days later, the small
start-up company WebExodus, which had been dragging its feet on making a
few final decisions, calls up Raj and offers him a summer job as well. Now, Raj
actually prefers WebExodus to CluNet—won over perhaps by the laid-back,
anything-can-happen atmosphere—and so this new development may well
cause him to retract his acceptance of the CluNet offer and go to WebExodus
instead. Suddenly down one summer intern, CluNet offers a job to one of its
wait-listed applicants, who promptly retracts his previous acceptance of an
offer from the software giant Babelsoft, and the situation begins to spiral out
of control.
Things look just as bad, if not worse, from the other direction. Suppose
that Raj’s friend Chelsea, destined to go to Babelsoft but having just heard Raj’s
story, calls up the people at WebExodus and says, “You know, I’d really rather
spend the summer with you guys than at Babelsoft.” They find this very easy
to believe; and furthermore, on looking at Chelsea’s application, they realize
that they would have rather hired her than some other student who actually
is scheduled to spend the summer at WebExodus. In this case, if WebExodus
were a slightly less scrupulous company, it might well find some way to retract
its offer to this other student and hire Chelsea instead.
Situations like this can rapidly generate a lot of chaos, and many people—
both applicants and employers—can end up unhappy with the process as well
as the outcome. What has gone wrong? One basic problem is that the process
is not self-enforcing—if people are allowed to act in their self-interest, then it
risks breaking down.
We might well prefer the following, more stable situation, in which selfinterest itself prevents offers from being retracted and redirected. Consider
another student, who has arranged to spend the summer at CluNet but calls
up WebExodus and reveals that he, too, would rather work for them. But in
this case, based on the offers already accepted, they are able to reply, “No, it
turns out that we prefer each of the students we’ve accepted to you, so we’re
afraid there’s nothing we can do.” Or consider an employer, earnestly following
up with its top applicants who went elsewhere, being told by each of them,
“No, I’m happy where I am.” In such a case, all the outcomes are stable—there
are no further outside deals that can be made.
So this is the question Gale and Shapley asked: Given a set of preferences
among employers and applicants, can we assign applicants to employers so
that for every employer E, and every applicant A who is not scheduled to work
for E, at least one of the following two things is the case?
1.1 A First Problem: Stable Matching
(i) E prefers every one of its accepted applicants to A; or
(ii) A prefers her current situation over working for employer E.
If this holds, the outcome is stable: individual self-interest will prevent any
applicant/employer deal from being made behind the scenes.
Gale and Shapley proceeded to develop a striking algorithmic solution to
this problem, which we will discuss presently. Before doing this, let’s note that
this is not the only origin of the Stable Matching Problem. It turns out that for
a decade before the work of Gale and Shapley, unbeknownst to them, the
National Resident Matching Program had been using a very similar procedure,
with the same underlying motivation, to match residents to hospitals. Indeed,
this system, with relatively little change, is still in use today.
This is one testament to the problem’s fundamental appeal. And from the
point of view of this book, it provides us with a nice first domain in which
to reason about some basic combinatorial definitions and the algorithms that
build on them.
Formulating the Problem To get at the essence of this concept, it helps to
make the problem as clean as possible. The world of companies and applicants
contains some distracting asymmetries. Each applicant is looking for a single
company, but each company is looking for many applicants; moreover, there
may be more (or, as is sometimes the case, fewer) applicants than there are
available slots for summer jobs. Finally, each applicant does not typically apply
to every company.
It is useful, at least initially, to eliminate these complications and arrive at a
more “bare-bones” version of the problem: each of n applicants applies to each
of n companies, and each company wants to accept a single applicant. We will
see that doing this preserves the fundamental issues inherent in the problem;
in particular, our solution to this simplified version will extend directly to the
more general case as well.
Following Gale and Shapley, we observe that this special case can be
viewed as the problem of devising a system by which each of n men and
n women can end up getting married: our problem naturally has the analogue
of two “genders”—the applicants and the companies—and in the case we are
considering, everyone is seeking to be paired with exactly one individual of
the opposite gender.1
Gale and Shapley considered the same-sex Stable Matching Problem as well, where there is only a
single gender. This is motivated by related applications, but it turns out to be fairly different at a
technical level. Given the applicant-employer application we’re considering here, we’ll be focusing
on the version with two genders.
Chapter 1 Introduction: Some Representative Problems
So consider a set M = {m1, . . . , mn } of n men, and a set W = {w1, . . . , wn }
of n women. Let M × W denote the set of all possible ordered pairs of the form
(m, w), where m ∈ M and w ∈ W. A matching S is a set of ordered pairs, each
from M × W, with the property that each member of M and each member of
W appears in at most one pair in S. A perfect matching S is a matching with
the property that each member of M and each member of W appears in exactly
one pair in S .
An instability: m and w⬘
each prefer the other to
their current partners.
Figure 1.1 Perfect matching
S with instability (m, w ).
Matchings and perfect matchings are objects that will recur frequently
throughout the book; they arise naturally in modeling a wide range of algorithmic problems. In the present situation, a perfect matching corresponds
simply to a way of pairing off the men with the women, in such a way that
everyone ends up married to somebody, and nobody is married to more than
one person—there is neither singlehood nor polygamy.
Now we can add the notion of preferences to this setting. Each man m ∈ M
ranks all the women; we will say that m prefers w to w if m ranks w higher
than w . We will refer to the ordered ranking of m as his preference list. We will
not allow ties in the ranking. Each woman, analogously, ranks all the men.
Given a perfect matching S, what can go wrong? Guided by our initial
motivation in terms of employers and applicants, we should be worried about
the following situation: There are two pairs (m, w) and (m , w ) in S (as
depicted in Figure 1.1) with the property that m prefers w to w, and w prefers
m to m . In this case, there’s nothing to stop m and w from abandoning their
current partners and heading off together; the set of marriages is not selfenforcing. We’ll say that such a pair (m, w ) is an instability with respect to S:
(m, w ) does not belong to S, but each of m and w prefers the other to their
partner in S.
Our goal, then, is a set of marriages with no instabilities. We’ll say that
a matching S is stable if (i) it is perfect, and (ii) there is no instability with
respect to S. Two questions spring immediately to mind:
Does there exist a stable matching for every set of preference lists?
Given a set of preference lists, can we efficiently construct a stable
matching if there is one?
Some Examples To illustrate these definitions, consider the following two
very simple instances of the Stable Matching Problem.
First, suppose we have a set of two men, {m, m }, and a set of two women,
{w, w }. The preference lists are as follows:
m prefers w to w .
m prefers w to w .
1.1 A First Problem: Stable Matching
w prefers m to m .
w prefers m to m .
If we think about this set of preference lists intuitively, it represents complete
agreement: the men agree on the order of the women, and the women agree
on the order of the men. There is a unique stable matching here, consisting
of the pairs (m, w) and (m , w ). The other perfect matching, consisting of the
pairs (m , w) and (m, w ), would not be a stable matching, because the pair
(m, w) would form an instability with respect to this matching. (Both m and
w would want to leave their respective partners and pair up.)
Next, here’s an example where things are a bit more intricate. Suppose
the preferences are
m prefers w to w .
m prefers w to w.
w prefers m to m.
w prefers m to m .
What’s going on in this case? The two men’s preferences mesh perfectly with
each other (they rank different women first), and the two women’s preferences
likewise mesh perfectly with each other. But the men’s preferences clash
completely with the women’s preferences.
In this second example, there are two different stable matchings. The
matching consisting of the pairs (m, w) and (m , w ) is stable, because both
men are as happy as possible, so neither would leave their matched partner.
But the matching consisting of the pairs (m , w) and (m, w ) is also stable, for
the complementary reason that both women are as happy as possible. This is
an important point to remember as we go forward—it’s possible for an instance
to have more than one stable matching.
Designing the Algorithm
We now show that there exists a stable matching for every set of preference
lists among the men and women. Moreover, our means of showing this will
also answer the second question that we asked above: we will give an efficient
algorithm that takes the preference lists and constructs a stable matching.
Let us consider some of the basic ideas that motivate the algorithm.
Initially, everyone is unmarried. Suppose an unmarried man m chooses
the woman w who ranks highest on his preference list and proposes to
her. Can we declare immediately that (m, w) will be one of the pairs in our
final stable matching? Not necessarily: at some point in the future, a man
m whom w prefers may propose to her. On the other hand, it would be
Chapter 1 Introduction: Some Representative Problems
dangerous for w to reject m right away; she may never receive a proposal
from someone she ranks as highly as m. So a natural idea would be to
have the pair (m, w) enter an intermediate state—engagement.
Woman w will become
engaged to m if she
prefers him to m⬘.
Figure 1.2 An intermediate
state of the G-S algorithm
when a free man m is proposing to a woman w.
Suppose we are now at a state in which some men and women are free—
not engaged—and some are engaged. The next step could look like this.
An arbitrary free man m chooses the highest-ranked woman w to whom
he has not yet proposed, and he proposes to her. If w is also free, then m
and w become engaged. Otherwise, w is already engaged to some other
man m . In this case, she determines which of m or m ranks higher
on her preference list; this man becomes engaged to w and the other
becomes free.
Finally, the algorithm will terminate when no one is free; at this moment,
all engagements are declared final, and the resulting perfect matching is
Here is a concrete description of the Gale-Shapley algorithm, with Figure 1.2 depicting a state of the algorithm.
Initially all m ∈ M and w ∈ W are free
While there is a man m who is free and hasn’t proposed to
every woman
Choose such a man m
Let w be the highest-ranked woman in m’s preference list
to whom m has not yet proposed
If w is free then
(m, w) become engaged
Else w is currently engaged to m
If w prefers m to m then
m remains free
Else w prefers m to m
(m, w) become engaged
m becomes free
Return the set S of engaged pairs
An intriguing thing is that, although the G-S algorithm is quite simple
to state, it is not immediately obvious that it returns a stable matching, or
even a perfect matching. We proceed to prove this now, through a sequence
of intermediate facts.
1.1 A First Problem: Stable Matching
Analyzing the Algorithm
First consider the view of a woman w during the execution of the algorithm.
For a while, no one has proposed to her, and she is free. Then a man m may
propose to her, and she becomes engaged. As time goes on, she may receive
additional proposals, accepting those that increase the rank of her partner. So
we discover the following.
(1.1) w remains engaged from the point at which she receives her first
proposal; and the sequence of partners to which she is engaged gets better and
better (in terms of her preference list).
The view of a man m during the execution of the algorithm is rather
different. He is free until he proposes to the highest-ranked woman on his
list; at this point he may or may not become engaged. As time goes on, he
may alternate between being free and being engaged; however, the following
property does hold.
(1.2) The sequence of women to whom m proposes gets worse and worse (in
terms of his preference list).
Now we show that the algorithm terminates, and give a bound on the
maximum number of iterations needed for termination.
The G-S algorithm terminates after at most n2 iterations of the While
Proof. A useful strategy for upper-bounding the running time of an algorithm,
as we are trying to do here, is to find a measure of progress. Namely, we seek
some precise way of saying that each step taken by the algorithm brings it
closer to termination.
In the case of the present algorithm, each iteration consists of some man
proposing (for the only time) to a woman he has never proposed to before. So
if we let P(t) denote the set of pairs (m, w) such that m has proposed to w by
the end of iteration t, we see that for all t, the size of P(t + 1) is strictly greater
than the size of P(t). But there are only n2 possible pairs of men and women
in total, so the value of P(·) can increase at most n2 times over the course of
the algorithm. It follows that there can be at most n2 iterations.
Two points are worth noting about the previous fact and its proof. First,
there are executions of the algorithm (with certain preference lists) that can
involve close to n2 iterations, so this analysis is not far from the best possible.
Second, there are many quantities that would not have worked well as a
progress measure for the algorithm, since they need not strictly increase in each
Chapter 1 Introduction: Some Representative Problems
iteration. For example, the number of free individuals could remain constant
from one iteration to the next, as could the number of engaged pairs. Thus,
these quantities could not be used directly in giving an upper bound on the
maximum possible number of iterations, in the style of the previous paragraph.
Let us now establish that the set S returned at the termination of the
algorithm is in fact a perfect matching. Why is this not immediately obvious?
Essentially, we have to show that no man can “fall off” the end of his preference
list; the only way for the While loop to exit is for there to be no free man. In
this case, the set of engaged couples would indeed be a perfect matching.
So the main thing we need to show is the following.
(1.4) If m is free at some point in the execution of the algorithm, then there
is a woman to whom he has not yet proposed.
Proof. Suppose there comes a point when m is free but has already proposed
to every woman. Then by (1.1), each of the n women is engaged at this point
in time. Since the set of engaged pairs forms a matching, there must also be
n engaged men at this point in time. But there are only n men total, and m is
not engaged, so this is a contradiction.
The set S returned at termination is a perfect matching.
Proof. The set of engaged pairs always forms a matching. Let us suppose that
the algorithm terminates with a free man m. At termination, it must be the
case that m had already proposed to every woman, for otherwise the While
loop would not have exited. But this contradicts (1.4), which says that there
cannot be a free man who has proposed to every woman.
Finally, we prove the main property of the algorithm—namely, that it
results in a stable matching.
(1.6) Consider an execution of the G-S algorithm that returns a set of pairs
S. The set S is a stable matching.
Proof. We have already seen, in (1.5), that S is a perfect matching. Thus, to
prove S is a stable matching, we will assume that there is an instability with
respect to S and obtain a contradiction. As defined earlier, such an instability
would involve two pairs, (m, w) and (m , w ), in S with the properties that
m prefers w to w, and
w prefers m to m .
In the execution of the algorithm that produced S, m’s last proposal was, by
definition, to w. Now we ask: Did m propose to w at some earlier point in
1.1 A First Problem: Stable Matching
this execution? If he didn’t, then w must occur higher on m’s preference list
than w , contradicting our assumption that m prefers w to w. If he did, then
he was rejected by w in favor of some other man m , whom w prefers to m.
m is the final partner of w , so either m = m or, by (1.1), w prefers her final
partner m to m ; either way this contradicts our assumption that w prefers
m to m .
It follows that S is a stable matching.
We began by defining the notion of a stable matching; we have just proven
that the G-S algorithm actually constructs one. We now consider some further
questions about the behavior of the G-S algorithm and its relation to the
properties of different stable matchings.
To begin with, recall that we saw an example earlier in which there could
be multiple stable matchings. To recap, the preference lists in this example
were as follows:
m prefers w to w .
m prefers w to w.
w prefers m to m.
w prefers m to m .
Now, in any execution of the Gale-Shapley algorithm, m will become engaged
to w, m will become engaged to w (perhaps in the other order), and things
will stop there. Thus, the other stable matching, consisting of the pairs (m , w)
and (m, w ), is not attainable from an execution of the G-S algorithm in which
the men propose. On the other hand, it would be reached if we ran a version of
the algorithm in which the women propose. And in larger examples, with more
than two people on each side, we can have an even larger collection of possible
stable matchings, many of them not achievable by any natural algorithm.
This example shows a certain “unfairness” in the G-S algorithm, favoring
men. If the men’s preferences mesh perfectly (they all list different women as
their first choice), then in all runs of the G-S algorithm all men end up matched
with their first choice, independent of the preferences of the women. If the
women’s preferences clash completely with the men’s preferences (as was the
case in this example), then the resulting stable matching is as bad as possible
for the women. So this simple set of preference lists compactly summarizes a
world in which someone is destined to end up unhappy: women are unhappy
if men propose, and men are unhappy if women propose.
Let’s now analyze the G-S algorithm in more detail and try to understand
how general this “unfairness” phenomenon is.
Chapter 1 Introduction: Some Representative Problems
To begin with, our example reinforces the point that the G-S algorithm
is actually underspecified: as long as there is a free man, we are allowed to
choose any free man to make the next proposal. Different choices specify
different executions of the algorithm; this is why, to be careful, we stated (1.6)
as “Consider an execution of the G-S algorithm that returns a set of pairs S,”
instead of “Consider the set S returned by the G-S algorithm.”
Thus, we encounter another very natural question: Do all executions of
the G-S algorithm yield the same matching? This is a genre of question that
arises in many settings in computer science: we have an algorithm that runs
asynchronously, with different independent components performing actions
that can be interleaved in complex ways, and we want to know how much
variability this asynchrony causes in the final outcome. To consider a very
different kind of example, the independent components may not be men and
women but electronic components activating parts of an airplane wing; the
effect of asynchrony in their behavior can be a big deal.
In the present context, we will see that the answer to our question is
surprisingly clean: all executions of the G-S algorithm yield the same matching.
We proceed to prove this now.
All Executions Yield the Same Matching There are a number of possible
ways to prove a statement such as this, many of which would result in quite
complicated arguments. It turns out that the easiest and most informative approach for us will be to uniquely characterize the matching that is obtained and
then show that all executions result in the matching with this characterization.
What is the characterization? We’ll show that each man ends up with the
“best possible partner” in a concrete sense. (Recall that this is true if all men
prefer different women.) First, we will say that a woman w is a valid partner
of a man m if there is a stable matching that contains the pair (m, w). We will
say that w is the best valid partner of m if w is a valid partner of m, and no
woman whom m ranks higher than w is a valid partner of his. We will use
best(m) to denote the best valid partner of m.
Now, let S∗ denote the set of pairs {(m, best(m)) : m ∈ M}. We will prove
the following fact.
Every execution of the G-S algorithm results in the set S∗.
This statement is surprising at a number of levels. First of all, as defined,
there is no reason to believe that S∗ is a matching at all, let alone a stable
matching. After all, why couldn’t it happen that two men have the same best
valid partner? Second, the result shows that the G-S algorithm gives the best
possible outcome for every man simultaneously; there is no stable matching
in which any of the men could have hoped to do better. And finally, it answers
1.1 A First Problem: Stable Matching
our question above by showing that the order of proposals in the G-S algorithm
has absolutely no effect on the final outcome.
Despite all this, the proof is not so difficult.
Proof. Let us suppose, by way of contradiction, that some execution E of the
G-S algorithm results in a matching S in which some man is paired with a
woman who is not his best valid partner. Since men propose in decreasing
order of preference, this means that some man is rejected by a valid partner
during the execution E of the algorithm. So consider the first moment during
the execution E in which some man, say m, is rejected by a valid partner w.
Again, since men propose in decreasing order of preference, and since this is
the first time such a rejection has occurred, it must be that w is m’s best valid
partner best(m).
The rejection of m by w may have happened either because m proposed
and was turned down in favor of w’s existing engagement, or because w broke
her engagement to m in favor of a better proposal. But either way, at this
moment w forms or continues an engagement with a man m whom she prefers
to m.
Since w is a valid partner of m, there exists a stable matching S containing
the pair (m, w). Now we ask: Who is m paired with in this matching? Suppose
it is a woman w = w.
Since the rejection of m by w was the first rejection of a man by a valid
partner in the execution E, it must be that m had not been rejected by any valid
partner at the point in E when he became engaged to w. Since he proposed in
decreasing order of preference, and since w is clearly a valid partner of m , it
must be that m prefers w to w . But we have already seen that w prefers m
to m, for in execution E she rejected m in favor of m . Since (m , w) ∈ S , it
follows that (m , w) is an instability in S .
This contradicts our claim that S is stable and hence contradicts our initial
So for the men, the G-S algorithm is ideal. Unfortunately, the same cannot
be said for the women. For a woman w, we say that m is a valid partner if
there is a stable matching that contains the pair (m, w). We say that m is the
worst valid partner of w if m is a valid partner of w, and no man whom w
ranks lower than m is a valid partner of hers.
(1.8) In the stable matching S∗, each woman is paired with her worst valid
Proof. Suppose there were a pair (m, w) in S∗ such that m is not the worst
valid partner of w. Then there is a stable matching S in which w is paired
Chapter 1 Introduction: Some Representative Problems
with a man m whom she likes less than m. In S , m is paired with a woman
w = w; since w is the best valid partner of m, and w is a valid partner of m,
we see that m prefers w to w .
But from this it follows that (m, w) is an instability in S , contradicting the
claim that S is stable and hence contradicting our initial assumption.
Thus, we find that our simple example above, in which the men’s preferences clashed with the women’s, hinted at a very general phenomenon: for
any input, the side that does the proposing in the G-S algorithm ends up with
the best possible stable matching (from their perspective), while the side that
does not do the proposing correspondingly ends up with the worst possible
stable matching.
1.2 Five Representative Problems
The Stable Matching Problem provides us with a rich example of the process of
algorithm design. For many problems, this process involves a few significant
steps: formulating the problem with enough mathematical precision that we
can ask a concrete question and start thinking about algorithms to solve
it; designing an algorithm for the problem; and analyzing the algorithm by
proving it is correct and giving a bound on the running time so as to establish
the algorithm’s efficiency.
This high-level strategy is carried out in practice with the help of a few
fundamental design techniques, which are very useful in assessing the inherent
complexity of a problem and in formulating an algorithm to solve it. As in any
area, becoming familiar with these design techniques is a gradual process; but
with experience one can start recognizing problems as belonging to identifiable
genres and appreciating how subtle changes in the statement of a problem can
have an enormous effect on its computational difficulty.
To get this discussion started, then, it helps to pick out a few representative milestones that we’ll be encountering in our study of algorithms: cleanly
formulated problems, all resembling one another at a general level, but differing greatly in their difficulty and in the kinds of approaches that one brings
to bear on them. The first three will be solvable efficiently by a sequence of
increasingly subtle algorithmic techniques; the fourth marks a major turning
point in our discussion, serving as an example of a problem believed to be unsolvable by any efficient algorithm; and the fifth hints at a class of problems
believed to be harder still.
The problems are self-contained and are all motivated by computing
applications. To talk about some of them, though, it will help to use the
terminology of graphs. While graphs are a common topic in earlier computer
1.2 Five Representative Problems
science courses, we’ll be introducing them in a fair amount of depth in
Chapter 3; due to their enormous expressive power, we’ll also be using them
extensively throughout the book. For the discussion here, it’s enough to think
of a graph G as simply a way of encoding pairwise relationships among a set
of objects. Thus, G consists of a pair of sets (V , E)—a collection V of nodes
and a collection E of edges, each of which “joins” two of the nodes. We thus
represent an edge e ∈ E as a two-element subset of V: e = {u, v} for some
u, v ∈ V, where we call u and v the ends of e. We typically draw graphs as in
Figure 1.3, with each node as a small circle and each edge as a line segment
joining its two ends.
Let’s now turn to a discussion of the five representative problems.
Interval Scheduling
Consider the following very simple scheduling problem. You have a resource—
it may be a lecture room, a supercomputer, or an electron microscope—and
many people request to use the resource for periods of time. A request takes
the form: Can I reserve the resource starting at time s, until time f ? We will
assume that the resource can be used by at most one person at a time. A
scheduler wants to accept a subset of these requests, rejecting all others, so
that the accepted requests do not overlap in time. The goal is to maximize the
number of requests accepted.
More formally, there will be n requests labeled 1, . . . , n, with each request
i specifying a start time si and a finish time fi . Naturally, we have si < fi for all i. Two requests i and j are compatible if the requested intervals do not overlap: that is, either request i is for an earlier time interval than request j (fi ≤ sj ), or request i is for a later time than request j (fj ≤ si ). We’ll say more generally that a subset A of requests is compatible if all pairs of requests i, j ∈ A, i = j are compatible. The goal is to select a compatible subset of requests of maximum possible size. We illustrate an instance of this Interval Scheduling Problem in Figure 1.4. Note that there is a single compatible set of size 4, and this is the largest compatible set. Figure 1.4 An instance of the Interval Scheduling Problem. (b) Figure 1.3 Each of (a) and (b) depicts a graph on four nodes. 14 Chapter 1 Introduction: Some Representative Problems We will see shortly that this problem can be solved by a very natural algorithm that orders the set of requests according to a certain heuristic and then “greedily” processes them in one pass, selecting as large a compatible subset as it can. This will be typical of a class of greedy algorithms that we will consider for various problems—myopic rules that process the input one piece at a time with no apparent look-ahead. When a greedy algorithm can be shown to find an optimal solution for all instances of a problem, it’s often fairly surprising. We typically learn something about the structure of the underlying problem from the fact that such a simple approach can be optimal. Weighted Interval Scheduling In the Interval Scheduling Problem, we sought to maximize the number of requests that could be accommodated simultaneously. Now, suppose more generally that each request interval i has an associated value, or weight, vi > 0; we could picture this as the amount of money we will make from
the ith individual if we schedule his or her request. Our goal will be to find a
compatible subset of intervals of maximum total value.
The case in which vi = 1 for each i is simply the basic Interval Scheduling
Problem; but the appearance of arbitrary values changes the nature of the
maximization problem quite a bit. Consider, for example, that if v1 exceeds
the sum of all other vi , then the optimal solution must include interval 1
regardless of the configuration of the full set of intervals. So any algorithm
for this problem must be very sensitive to the values, and yet degenerate to a
method for solving (unweighted) interval scheduling when all the values are
equal to 1.
There appears to be no simple greedy rule that walks through the intervals
one at a time, making the correct decision in the presence of arbitrary values.
Instead, we employ a technique, dynamic programming, that builds up the
optimal value over all possible solutions in a compact, tabular way that leads
to a very efficient algorithm.
Bipartite Matching
When we considered the Stable Matching Problem, we defined a matching to
be a set of ordered pairs of men and women with the property that each man
and each woman belong to at most one of the ordered pairs. We then defined
a perfect matching to be a matching in which every man and every woman
belong to some pair.
We can express these concepts more generally in terms of graphs, and in
order to do this it is useful to define the notion of a bipartite graph. We say that
a graph G = (V , E) is bipartite if its node set V can be partitioned into sets X
1.2 Five Representative Problems
and Y in such a way that every edge has one end in X and the other end in Y.
A bipartite graph is pictured in Figure 1.5; often, when we want to emphasize
a graph’s “bipartiteness,” we will draw it this way, with the nodes in X and
Y in two parallel columns. But notice, for example, that the two graphs in
Figure 1.3 are also bipartite.
Now, in the problem of finding a stable matching, matchings were built
from pairs of men and women. In the case of bipartite graphs, the edges are
pairs of nodes, so we say that a matching in a graph G = (V , E) is a set of edges
M ⊆ E with the property that each node appears in at most one edge of M.
M is a perfect matching if every node appears in exactly one edge of M.
To see that this does capture the same notion we encountered in the Stable
Matching Problem, consider a bipartite graph G with a set X of n men, a set Y
of n women, and an edge from every node in X to every node in Y. Then the
matchings and perfect matchings in G are precisely the matchings and perfect
matchings among the set of men and women.
In the Stable Matching Problem, we added preferences to this picture. Here,
we do not consider preferences; but the nature of the problem in arbitrary
bipartite graphs adds a different source of complexity: there is not necessarily
an edge from every x ∈ X to every y ∈ Y, so the set of possible matchings has
quite a complicated structure. In other words, it is as though only certain pairs
of men and women are willing to be paired off, and we want to figure out
how to pair off many people in a way that is consistent with this. Consider,
for example, the bipartite graph G in Figure 1.5: there are many matchings in
G, but there is only one perfect matching. (Do you see it?)
Matchings in bipartite graphs can model situations in which objects are
being assigned to other objects. Thus, the nodes in X can represent jobs, the
nodes in Y can represent machines, and an edge (xi , yj ) can indicate that
machine yj is capable of processing job xi . A perfect matching is then a way
of assigning each job to a machine that can process it, with the property that
each machine is assigned exactly one job. In the spring, computer science
departments across the country are often seen pondering a bipartite graph in
which X is the set of professors in the department, Y is the set of offered
courses, and an edge (xi , yj ) indicates that professor xi is capable of teaching
course yj . A perfect matching in this graph consists of an assignment of each
professor to a course that he or she can teach, in such a way that every course
is covered.
Thus the Bipartite Matching Problem is the following: Given an arbitrary
bipartite graph G, find a matching of maximum size. If |X| = |Y| = n, then there
is a perfect matching if and only if the maximum matching has size n. We will
find that the algorithmic techniques discussed earlier do not seem adequate
Figure 1.5 A bipartite graph.
Chapter 1 Introduction: Some Representative Problems
for providing an efficient algorithm for this problem. There is, however, a very
elegant and efficient algorithm to find a maximum matching; it inductively
builds up larger and larger matchings, selectively backtracking along the way.
This process is called augmentation, and it forms the central component in a
large class of efficiently solvable problems called network flow problems.
Independent Set
Figure 1.6 A graph whose
largest independent set has
size 4.
Now let’s talk about an extremely general problem, which includes most of
these earlier problems as special cases. Given a graph G = (V , E), we say
a set of nodes S ⊆ V is independent if no two nodes in S are joined by an
edge. The Independent Set Problem is, then, the following: Given G, find an
independent set that is as large as possible. For example, the maximum size of
an independent set in the graph in Figure 1.6 is four, achieved by the four-node
independent set {1, 4, 5, 6}.
The Independent Set Problem encodes any situation in which you are
trying to choose from among a collection of objects and there are pairwise
conflicts among some of the objects. Say you have n friends, and some pairs
of them don’t get along. How large a group of your friends can you invite to
dinner if you don’t want any interpersonal tensions? This is simply the largest
independent set in the graph whose nodes are your friends, with an edge
between each conflicting pair.
Interval Scheduling and Bipartite Matching can both be encoded as special
cases of the Independent Set Problem. For Interval Scheduling, define a graph
G = (V , E) in which the nodes are the intervals and there is an edge between
each pair of them that overlap; the independent sets in G are then just the
compatible subsets of intervals. Encoding Bipartite Matching as a special case
of Independent Set is a little trickier to see. Given a bipartite graph G = (V , E ),
the objects being chosen are edges, and the conflicts arise between two edges
that share an end. (These, indeed, are the pairs of edges that cannot belong
to a common matching.) So we define a graph G = (V , E) in which the node
set V is equal to the edge set E of G . We define an edge between each pair
of elements in V that correspond to edges of G with a common end. We can
now check that the independent sets of G are precisely the matchings of G .
While it is not complicated to check this, it takes a little concentration to deal
with this type of “edges-to-nodes, nodes-to-edges” transformation.2
For those who are curious, we note that not every instance of the Independent Set Problem can arise
in this way from Interval Scheduling or from Bipartite Matching; the full Independent Set Problem
really is more general. The graph in Figure 1.3(a) cannot arise as the “conflict graph” in an instance of
1.2 Five Representative Problems
Given the generality of the Independent Set Problem, an efficient algorithm
to solve it would be quite impressive. It would have to implicitly contain
algorithms for Interval Scheduling, Bipartite Matching, and a host of other
natural optimization problems.
The current status of Independent Set is this: no efficient algorithm is
known for the problem, and it is conjectured that no such algorithm exists.
The obvious brute-force algorithm would try all subsets of the nodes, checking
each to see if it is independent, and then recording the largest one encountered.
It is possible that this is close to the best we can do on this problem. We will
see later in the book that Independent Set is one of a large class of problems
that are termed NP-complete. No efficient algorithm is known for any of them;
but they are all equivalent in the sense that a solution to any one of them
would imply, in a precise sense, a solution to all of them.
Here’s a natural question: Is there anything good we can say about the
complexity of the Independent Set Problem? One positive thing is the following:
If we have a graph G on 1,000 nodes, and we want to convince you that it
contains an independent set S of size 100, then it’s quite easy. We simply
show you the graph G, circle the nodes of S in red, and let you check that
no two of them are joined by an edge. So there really seems to be a great
difference in difficulty between checking that something is a large independent
set and actually finding a large independent set. This may look like a very basic
observation—and it is—but it turns out to be crucial in understanding this class
of problems. Furthermore, as we’ll see next, it’s possible for a problem to be
so hard that there isn’t even an easy way to “check” solutions in this sense.
Competitive Facility Location
Finally, we come to our fifth problem, which is based on the following twoplayer game. Consider two large companies that operate café franchises across
the country—let’s call them JavaPlanet and Queequeg’s Coffee—and they are
currently competing for market share in a geographic area. First JavaPlanet
opens a franchise; then Queequeg’s Coffee opens a franchise; then JavaPlanet;
then Queequeg’s; and so on. Suppose they must deal with zoning regulations
that require no two franchises be located too close together, and each is trying
to make its locations as convenient as possible. Who will win?
Let’s make the rules of this “game” more concrete. The geographic region
in question is divided into n zones, labeled 1, 2, . . . , n. Each zone i has a
Interval Scheduling, and the graph in Figure 1.3(b) cannot arise as the “conflict graph” in an instance
of Bipartite Matching.
Chapter 1 Introduction: Some Representative Problems
Figure 1.7 An instance of the Competitive Facility Location Problem.
value bi , which is the revenue obtained by either of the companies if it opens
a franchise there. Finally, certain pairs of zones (i, j) are adjacent, and local
zoning laws prevent two adjacent zones from each containing a franchise,
regardless of which company owns them. (They also prevent two franchises
from being opened in the same zone.) We model these conflicts via a graph
G = (V , E), where V is the set of zones, and (i, j) is an edge in E if the
zones i and j are adjacent. The zoning requirement then says that the full
set of franchises opened must form an independent set in G.
Thus our game consists of two players, P1 and P2, alternately selecting
nodes in G, with P1 moving first. At all times, the set of all selected nodes
must form an independent set in G. Suppose that player P2 has a target bound
B, and we want to know: is there a strategy for P2 so that no matter how P1
plays, P2 will be able to select a set of nodes with a total value of at least B?
We will call this an instance of the Competitive Facility Location Problem.
Consider, for example, the instance pictured in Figure 1.7, and suppose
that P2’s target bound is B = 20. Then P2 does have a winning strategy. On the
other hand, if B = 25, then P2 does not.
One can work this out by looking at the figure for a while; but it requires
some amount of case-checking of the form, “If P1 goes here, then P2 will go
there; but if P1 goes over there, then P2 will go here. . . . ” And this appears to
be intrinsic to the problem: not only is it computationally difficult to determine
whether P2 has a winning strategy; on a reasonably sized graph, it would even
be hard for us to convince you that P2 has a winning strategy. There does not
seem to be a short proof we could present; rather, we’d have to lead you on a
lengthy case-by-case analysis of the set of possible moves.
This is in contrast to the Independent Set Problem, where we believe that
finding a large solution is hard but checking a proposed large solution is easy.
This contrast can be formalized in the class of PSPACE-complete problems, of
which Competitive Facility Location is an example. PSPACE-complete problems are believed to be strictly harder than NP-complete problems, and this
conjectured lack of short “proofs” for their solutions is one indication of this
greater hardness. The notion of PSPACE-completeness turns out to capture a
large collection of problems involving game-playing and planning; many of
these are fundamental issues in the area of artificial intelligence.
Solved Exercises
Solved Exercises
Solved Exercise 1
Consider a town with n men and n women seeking to get married to one
another. Each man has a preference list that ranks all the women, and each
woman has a preference list that ranks all the men.
The set of all 2n people is divided into two categories: good people and
bad people. Suppose that for some number k, 1 ≤ k ≤ n − 1, there are k good
men and k good women; thus there are n − k bad men and n − k bad women.
Everyone would rather marry any good person than any bad person.
Formally, each preference list has the property that it ranks each good person
of the opposite gender higher than each bad person of the opposite gender: its
first k entries are the good people (of the opposite gender) in some order, and
its next n − k are the bad people (of the opposite gender) in some order.
Show that in every stable matching, every good man is married to a good
Solution A natural way to get started thinking about this problem is to
assume the claim is false and try to work toward obtaining a contradiction.
What would it mean for the claim to be false? There would exist some stable
matching M in which a good man m was married to a bad woman w.
Now, let’s consider what the other pairs in M look like. There are k good
men and k good women. Could it be the case that every good woman is married
to a good man in this matching M? No: one of the good men (namely, m) is
already married to a bad woman, and that leaves only k − 1 other good men.
So even if all of them were married to good women, that would still leave some
good woman who is married to a bad man.
Let w be such a good woman, who is married to a bad man. It is now
easy to identify an instability in M: consider the pair (m, w ). Each is good,
but is married to a bad partner. Thus, each of m and w prefers the other to
their current partner, and hence (m, w ) is an instability. This contradicts our
assumption that M is stable, and hence concludes the proof.
Solved Exercise 2
We can think about a generalization of the Stable Matching Problem in which
certain man-woman pairs are explicitly forbidden. In the case of employers and
applicants, we could imagine that certain applicants simply lack the necessary
qualifications or certifications, and so they cannot be employed at certain
companies, however desirable they may seem. Using the analogy to marriage
between men and women, we have a set M of n men, a set W of n women,
Chapter 1 Introduction: Some Representative Problems
and a set F ⊆ M × W of pairs who are simply not allowed to get married. Each
man m ranks all the women w for which (m, w) ∈ F, and each woman w ranks
all the men m for which (m , w ) ∈ F.
In this more general setting, we say that a matching S is stable if it does
not exhibit any of the following types of instability.
(i) There are two pairs (m, w) and (m , w ) in S with the property that
(m, w ) ∈ F, m prefers w to w, and w prefers m to m . (The usual kind
of instability.)
(ii) There is a pair (m, w) ∈ S, and a man m , so that m is not part of any
pair in the matching, (m , w) ∈ F, and w prefers m to m. (A single man
is more desirable and not forbidden.)
(iii) There is a pair (m, w) ∈ S, and a woman w , so that w is not part of
any pair in the matching, (m, w ) ∈ F, and m prefers w to w. (A single
woman is more desirable and not forbidden.)
(iv) There is a man m and a woman w, neither of whom is part of any pair
in the matching, so that (m, w) ∈ F. (There are two single people with
nothing preventing them from getting married to each other.)
Note that under these more general definitions, a stable matching need not be
a perfect matching.
Now we can ask: For every set of preference lists and every set of forbidden
pairs, is there always a stable matching? Resolve this question by doing one of
the following two things: (a) give an algorithm that, for any set of preference
lists and forbidden pairs, produces a stable matching; or (b) give an example
of a set of preference lists and forbidden pairs for which there is no stable
Solution The Gale-Shapley algorithm is remarkably robust to variations on
the Stable Matching Problem. So, if you’re faced with a new variation of the
problem and can’t find a counterexample to stability, it’s often a good idea to
check whether a direct adaptation of the G-S algorithm will in fact produce
stable matchings.
That turns out to be the case here. We will show that there is always a
stable matching, even in this more general model with forbidden pairs, and
we will do this by adapting the G-S algorithm. To do this, let’s consider why
the original G-S algorithm can’t be used directly. The difficulty, of course, is
that the G-S algorithm doesn’t know anything about forbidden pairs, and so
the condition in the While loop,
While there is a man m who is free and hasn’t proposed to
every woman,
Solved Exercises
won’t work: we don’t want m to propose to a woman w for which the pair
(m, w) is forbidden.
Thus, let’s consider a variation of the G-S algorithm in which we make
only one change: we modify the While loop to say,
While there is a man m who is free and hasn’t proposed to
every woman w for which (m, w) ∈ F.
Here is the algorithm in full.
Initially all m ∈ M and w ∈ W are free
While there is a man m who is free and hasn’t proposed to
every woman w for which (m, w) ∈ F
Choose such a man m
Let w be the highest-ranked woman in m’s preference list
to which m has not yet proposed
If w is free then
(m, w) become engaged
Else w is currently engaged to m
If w prefers m to m then
m remains free
Else w prefers m to m
(m, w) become engaged
m becomes free
Return the set S of engaged pairs
We now prove that this yields a stable matching, under our new definition
of stability.
To begin with, facts (1.1), (1.2), and (1.3) from the text remain true (in
particular, the algorithm will terminate in at most n2 iterations). Also, we
don’t have to worry about establishing that the resulting matching S is perfect
(indeed, it may not be). We also notice an additional pairs of facts. If m is
a man who is not part of a pair in S, then m must have proposed to every
nonforbidden woman; and if w is a woman who is not part of a pair in S, then
it must be that no man ever proposed to w.
Finally, we need only show
There is no instability with respect to the returned matching S.
Chapter 1 Introduction: Some Representative Problems
Proof. Our general definition of instability has four parts: This means that we
have to make sure that none of the four bad things happens.
First, suppose there is an instability of type (i), consisting of pairs (m, w)
and (m , w ) in S with the property that (m, w ) ∈ F, m prefers w to w, and w
prefers m to m . It follows that m must have proposed to w ; so w rejected m,
and thus she prefers her final partner to m—a contradiction.
Next, suppose there is an instability of type (ii), consisting of a pair
(m, w) ∈ S, and a man m , so that m is not part of any pair in the matching,
(m , w) ∈ F, and w prefers m to m. Then m must have proposed to w and
been rejected; again, it follows that w prefers her final partner to m —a
Third, suppose there is an instability of type (iii), consisting of a pair
(m, w) ∈ S, and a woman w , so that w is not part of any pair in the matching,
(m, w ) ∈ F, and m prefers w to w. Then no man proposed to w at all;
in particular, m never proposed to w , and so he must prefer w to w —a
Finally, suppose there is an instability of type (iv), consisting of a man
m and a woman w, neither of which is part of any pair in the matching,
so that (m, w) ∈ F. But for m to be single, he must have proposed to every
nonforbidden woman; in particular, he must have proposed to w, which means
she would no longer be single—a contradiction.
1. Decide whether you think the following statement is true or false. If it is
true, give a short explanation. If it is false, give a counterexample.
True or false? In every instance of the Stable Matching Problem, there is a
stable matching containing a pair (m, w) such that m is ranked first on the
preference list of w and w is ranked first on the preference list of m.
2. Decide whether you think the following statement is true or false. If it is
true, give a short explanation. If it is false, give a counterexample.
True or false? Consider an instance of the Stable Matching Problem in which
there exists a man m and a woman w such that m is ranked first on the
preference list of w and w is ranked first on the preference list of m. Then in
every stable matching S for this instance, the pair (m, w) belongs to S.
3. There are many other settings in which we can ask questions related
to some type of “stability” principle. Here’s one, involving competition
between two enterprises.
Suppose we have two television networks, whom we’ll call A and B.
There are n prime-time programming slots, and each network has n TV
shows. Each network wants to devise a schedule—an assignment of each
show to a distinct slot—so as to attract as much market share as possible.
Here is the way we determine how well the two networks perform
relative to each other, given their schedules. Each show has a fixed rating,
which is based on the number of people who watched it last year; we’ll
assume that no two shows have exactly the same rating. A network wins a
given time slot if the show that it schedules for the time slot has a larger
rating than the show the other network schedules for that time slot. The
goal of each network is to win as many time slots as possible.
Suppose in the opening week of the fall season, Network A reveals a
schedule S and Network B reveals a schedule T. On the basis of this pair
of schedules, each network wins certain time slots, according to the rule
above. We’ll say that the pair of schedules (S, T) is stable if neither network
can unilaterally change its own schedule and win more time slots. That
is, there is no schedule S such that Network A wins more slots with the
pair (S , T) than it did with the pair (S, T); and symmetrically, there is no
schedule T such that Network B wins more slots with the pair (S, T ) than
it did with the pair (S, T).
The analogue of Gale and Shapley’s question for this kind of stability
is the following: For every set of TV shows and ratings, is there always
a stable pair of schedules? Resolve this question by doing one of the
following two things:
(a) give an algorithm that, for any set of TV shows and associated
ratings, produces a stable pair of schedules; or
(b) give an example of a set of TV shows and associated ratings for
which there is no stable pair of schedules.
4. Gale and Shapley published their paper on the Stable Matching Problem
in 1962; but a version of their algorithm had already been in use for
ten years by the National Resident Matching Program, for the problem of
assigning medical residents to hospitals.
Basically, the situation was the following. There were m hospitals,
each with a certain number of available positions for hiring residents.
There were n medical students graduating in a given year, each interested
in joining one of the hospitals. Each hospital had a ranking of the students
in order of preference, and each student had a ranking of the hospitals
in order of preference. We will assume that there were more students
graduating than there were slots available in the m hospitals.
Chapter 1 Introduction: Some Representative Problems
The interest, naturally, was in finding a way of assigning each student
to at most one hospital, in such a way that all available positions in all
hospitals were filled. (Since we are assuming a surplus of students, there
would be some students who do not get assigned to any hospital.)
We say that an assignment of students to hospitals is stable if neither
of the following situations arises.
First type of instability: There are students s and s , and a hospital h,
so that
– s is assigned to h, and
– s is assigned to no hospital, and
– h prefers s to s.
Second type of instability: There are students s and s , and hospitals
h and h , so that
– s is assigned to h, and
– s is assigned to h , and
– h prefers s to s, and
– s prefers h to h .
So we basically have the Stable Matching Problem, except that (i)
hospitals generally want more than one resident, and (ii) there is a surplus
of medical students.
Show that there is always a stable assignment of students to hospitals, and give an algorithm to find one.
5. The Stable Matching Problem, as discussed in the text, assumes that all
men and women have a fully ordered list of preferences. In this problem
we will consider a version of the problem in which men and women can be
indifferent between certain options. As before we have a set M of n men
and a set W of n women. Assume each man and each woman ranks the
members of the opposite gender, but now we allow ties in the ranking.
For example (with n = 4), a woman could say that m1 is ranked in first
place; second place is a tie between m2 and m3 (she has no preference
between them); and m4 is in last place. We will say that w prefers m to m
if m is ranked higher than m on her preference list (they are not tied).
With indifferences in the rankings, there could be two natural notions
for stability. And for each, we can ask about the existence of stable
matchings, as follows.
A strong instability in a perfect matching S consists of a man m and
a woman w, such that each of m and w prefers the other to their
partner in S. Does there always exist a perfect matching with no
strong instability? Either give an example of a set of men and women
with preference lists for which every perfect matching has a strong
instability; or give an algorithm that is guaranteed to find a perfect
matching with no strong instability.
(b) A weak instability in a perfect matching S consists of a man m and
a woman w, such that their partners in S are w and m ,…
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