LOW COEFFICIENT COMPLEXITY APPROXIMATIONS OF THE ONE DIMENSIONAL DISCRETE COSINE TRANSFORM

A major goal of exploration geophysics is to determine the structure of the Earth near the surface from measured seismic data. Ideally, one would like to determine the important material parameters (e.g., density, wave speed(s)) as functions of position. This is di cult for a number of reasons: insu cient understanding of the physics (we do not really know the correct form of the di erential equations governing wave motion in the subsurface appropriate for inverting seismic data); the ill-posed nature of multidimensional inverse problems for wave equations (even if we knew the correct form of the equations, the coe cients may not be uniquely determined); the very large size of seismic data sets; and numerous practical di culties in working with seismic data (inaccurate knowledge of the source, noise in the data, etc.). One approach used with some success is to extract from large seismic data sets more tractable data sets consisting of re ection and/or transmission traveltimes. An inverse problem is then formulated for which the measured data are these traveltimes. In seismic tomographic inversion, we attemined from seismic data with the corresponding traveltimes computed from a model of the subsurface. Surface seismic re ection tomography (Bishop et al. 3]) uses traveltimes along rays from surface shot locations down to re ecting interfaces and then back up to surface receiver locations; the goal is to determine both the position of the re ectors and the slowness eld (reciprocal of wave speed) above the re ectors. Because the Earth is often layered near the surface, these re ecting interfaces are often nearly horizontal. In cross-well seismic tomography, transmission traveltimes from shot locations in one well to receiver locations in another well are measured, and the goal is to determine the slowness eld between the wells. We restrict our attention here to surface re ection tomography; most of our conclusions concerning the recovery of the slowness eld carry over directly to the cross-well geometry. Surface re ection tomography is not a well posed problem, either for continuum for1

2 Cov( X , X ) = E[( X − µ X )( X − µ X )] = E[( X − µ X ) 2 ] = σ X
4. If X and Y are two r.v. but are not necessarily independent, then
Var ( X + Y ) = Var ( X ) + Var (Y ) + 2Cov ( X , Y )
E (Y ) = ∑ E (Y / X = x) Pr( X = xi )
i =1 l
Stated differently,
E (Y ) = E[ E (Y / X )]
The probability framework for statistical inference Conditional Variance: The variance of Y conditional on X is the variance of the conditional distribution of Y given X. • The conditional variance of Y given X is
Pr(X = x,Y = y) Pr(Y = y / X = x) = Pr(X = x)
The probability framework for statistical inference Conditional Expectation: The conditional expectation of Y given X is the mean of the conditional distribution of Y given X. • If Y takes on k values (y1, y2,…,yk), then the conditional mean of Y given X=x is

Algorithmic Game TheoryOver the last few years,there has been explosive growth in the research done at the in-terface of computer science,game theory,and economic theory,largely motivated by the emergence of the Internet.Algorithmic Game Theory develops the central ideas and results of this new and exciting area.More than40of the top researchers in thisfield have written chapters whose topics range from the foundations to the state of the art.This book contains an extensive treatment of algorithms for equilibria in games and markets,computational auctions and mechanism design,and the“price of anarchy,”as well as applications in networks,peer-to-peer systems, security,information markets,and more.This book will be of interest to students,researchers,and practitioners in theoretical computer science,economics,networking,artificial intelligence,operations research,and discrete mathematics.Noam Nisan is a Professor in the Department of Computer Science at The Hebrew Univer-sity of Jerusalem.His other books include Communication Complexity.Tim Roughgarden is an Assistant Professor in the Department of Computer Science at Stanford University.His other books include Selfish Routing and the Price of Anarchy.´Eva Tardos is a Professor in the Department of Computer Science at Cornell University. Her other books include Algorithm Design.Vijay V.Vazirani is a Professor in the College of Computing at the Georgia Institute of Technology.His other books include Approximation Algorithms.Algorithmic Game TheoryEdited byNoam NisanHebrew University of JerusalemTim RoughgardenStanford University´Eva TardosCornell UniversityVijay V.VaziraniGeorgia Institute of Technologycambridge university pressCambridge,New York,Melbourne,Madrid,Cape Town,Singapore,S˜a o Paulo,Delhi Cambridge University Press32Avenue of the Americas,New York,NY10013-2473,USAInformation on this title:/9780521872829C Noam Nisan,Tim Roughgarden,´Eva Tardos,Vijay V.Vazirani2007This publication is in copyright.Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.First published2007Printed in the United States of AmericaA catalog record for this book is available from the British Library.Library of Congress Cataloging in Publication DataAlgorithmic game theory/edited by Noam Nisan...[et al.];forewordby Christos Papadimitriou.p.cm.Includes index.ISBN-13:978-0-521-87282-9(hardback)ISBN-10:0-521-87282-0(hardback)1.Game theory.2.Algorithms.I.Nisan,Noam.II.Title.QA269.A432007519.3–dc222007014231ISBN978-0-521-87282-9hardbackCambridge University Press has no responsibility forthe persistence or accuracy of URLS for external orthird-party Internet Web sites referred to in this publicationand does not guarantee that any content on suchWeb sites is,or will remain,accurate or appropriate.ContentsForeword page xiii Preface xvii Contributors xixI Computing in Games1Basic Solution Concepts and Computational Issues3´Eva Tardos and Vijay V.Vazirani1.1Games,Old and New31.2Games,Strategies,Costs,and Payoffs91.3Basic Solution Concepts101.4Finding Equilibria and Learning in Games161.5Refinement of Nash:Games with Turns and Subgame Perfect Equilibrium181.6Nash Equilibrium without Full Information:Bayesian Games201.7Cooperative Games201.8Markets and Their Algorithmic Issues22Acknowledgments26 Bibliography26 Exercises26 2The Complexity of Finding Nash Equilibria29 Christos H.Papadimitriou2.1Introduction292.2Is the N ash Equilibrium Problem NP-Complete?312.3The Lemke–Howson Algorithm332.4The Class PPAD362.5Succinct Representations of Games392.6The Reduction412.7Correlated Equilibria452.8Concluding Remarks49Acknowledgment50 Bibliography50vvi contents3Equilibrium Computation for Two-Player Games in Strategicand Extensive Form53 Bernhard von Stengel3.1Introduction533.2Bimatrix Games and the Best Response Condition543.3Equilibria via Labeled Polytopes573.4The Lemke–Howson Algorithm613.5Integer Pivoting633.6Degenerate Games653.7Extensive Games and Their Strategic Form663.8Subgame Perfect Equilibria683.9Reduced Strategic Form693.10The Sequence Form703.11Computing Equilibria with the Sequence Form733.12Further Reading753.13Discussion and Open Problems75Bibliography76 Exercises77 4Learning,Regret Minimization,and Equilibria79 Avrim Blum and Yishay Mansour4.1Introduction794.2Model and Preliminaries814.3External Regret Minimization824.4Regret Minimization and Game Theory884.5Generic Reduction from External to Swap Regret924.6The Partial Information Model944.7On Convergence of Regret-Minimizing Strategies to NashEquilibrium in Routing Games964.8Notes99Bibliography99 Exercises101 5Combinatorial Algorithms for Market Equilibria103 Vijay V.Vazirani5.1Introduction1035.2Fisher’s Linear Case and the Eisenberg–Gale Convex Program1055.3Checking If Given Prices Are Equilibrium Prices1085.4Two Crucial Ingredients of the Algorithm1095.5The Primal-Dual Schema in the Enhanced Setting1095.6Tight Sets and the Invariant1115.7Balanced Flows1115.8The Main Algorithm1155.9Finding Tight Sets1175.10Running Time of the Algorithm1185.11The Linear Case of the Arrow–Debreu Model1215.12An Auction-Based Algorithm1225.13Resource Allocation Markets124contents vii 5.14Algorithm for Single-Source Multiple-Sink Markets126 5.15Discussion and Open Problems131 Bibliography132 Exercises133 6Computation of Market Equilibria by Convex Programming135 Bruno Codenotti and Kasturi Varadarajan6.1Introduction1356.2Fisher Model with Homogeneous Consumers1416.3Exchange Economies Satisfying WGS1426.4Specific Utility Functions1486.5Limitations1506.6Models with Production1526.7Bibliographic Notes155 Bibliography156 Exercises158 7Graphical Games159 Michael Kearns7.1Introduction1597.2Preliminaries1617.3Computing Nash Equilibria in Tree Graphical Games1647.4Graphical Games and Correlated Equilibria1697.5Graphical Exchange Economies1767.6Open Problems and Future Research1777.7Bibliographic Notes177 Acknowledgments179 Bibliography179 8Cryptography and Game Theory181 Yevgeniy Dodis and Tal Rabin8.1Cryptographic Notions and Settings1818.2Game Theory Notions and Settings1878.3Contrasting MPC and Games1898.4Cryptographic Influences on Game Theory1918.5Game Theoretic Influences on Cryptography1978.6Conclusions2028.7Notes203 Acknowledgments204 Bibliography204 II Algorithmic Mechanism Design9Introduction to Mechanism Design(for Computer Scientists)209 Noam Nisan9.1Introduction2099.2Social Choice2119.3Mechanisms with Money2169.4Implementation in Dominant Strategies222viii contents9.5Characterizations of Incentive Compatible Mechanisms2259.6Bayesian–Nash Implementation2339.7Further Models2389.8Notes239Acknowledgments240 Bibliography241 10Mechanism Design without Money243 James Schummer and Rakesh V.Vohra10.1Introduction24310.2Single-Peaked Preferences over Policies24410.3House Allocation Problem25310.4Stable Matchings25510.5Future Directions26210.6Notes and References263Bibliography264 Exercises264 11Combinatorial Auctions267 Liad Blumrosen and Noam Nisan11.1Introduction26711.2The Single-Minded Case27011.3Walrasian Equilibrium and the LP Relaxation27511.4Bidding Languages27911.5Iterative Auctions:The Query Model28311.6Communication Complexity28711.7Ascending Auctions28911.8Bibliographic Notes295Acknowledgments296 Bibliography296 Exercises298 12Computationally Efficient Approximation Mechanisms301 Ron Lavi12.1Introduction30112.2Single-Dimensional Domains:Job Scheduling30312.3Multidimensional Domains:Combinatorial Auctions31012.4Impossibilities of Dominant Strategy Implementability31712.5Alternative Solution Concepts32112.6Bibliographic Notes327Bibliography327 Exercises328 13Profit Maximization in Mechanism Design331 Jason D.Hartline and Anna R.Karlin13.1Introduction33113.2Bayesian Optimal Mechanism Design33513.3Prior-Free Approximations to the Optimal Mechanism33913.4Prior-Free Optimal Mechanism Design344contents ix13.5Frugality35013.6Conclusions and Other Research Directions35413.7Notes357Bibliography358 Exercises360 14Distributed Algorithmic Mechanism Design363 Joan Feigenbaum,Michael Schapira,and Scott Shenker14.1Introduction36314.2Two Examples of DAMD36614.3Interdomain Routing37014.4Conclusion and Open Problems37914.5Notes380Acknowledgments381 Bibliography381 Exercises383 15Cost Sharing385 Kamal Jain and Mohammad Mahdian15.1Cooperative Games and Cost Sharing38515.2Core of Cost-Sharing Games38715.3Group-Strategyproof Mechanisms and Cross-MonotonicCost-Sharing Schemes39115.4Cost Sharing via the Primal-Dual Schema39415.5Limitations of Cross-Monotonic Cost-Sharing Schemes40015.6The Shapley Value and the Nash Bargaining Solution40215.7Conclusion40515.8Notes406Acknowledgments408 Bibliography408 Exercises410 16Online Mechanisms411 David C.Parkes16.1Introduction41116.2Dynamic Environments and Online MD41316.3Single-Valued Online Domains41716.4Bayesian Implementation in Online Domains43116.5Conclusions43516.6Notes436Acknowledgments437 Bibliography437 Exercises439 III Quantifying the Inefficiency of Equilibria17Introduction to the Inefficiency of Equilibria443 Tim Roughgarden and´Eva Tardos17.1Introduction443x contents17.2Fundamental Network Examples44617.3Inefficiency of Equilibria as a Design Metric45417.4Notes456Bibliography457 Exercises459 18Routing Games461 Tim Roughgarden18.1Introduction46118.2Models and Examples46218.3Existence,Uniqueness,and Potential Functions46818.4The Price of Anarchy of Selfish Routing47218.5Reducing the Price of Anarchy47818.6Notes480Bibliography483 Exercises484 19Network Formation Games and the Potential Function Method487´Eva Tardos and Tom Wexler19.1Introduction48719.2The Local Connection Game48919.3Potential Games and a Global Connection Game49419.4Facility Location50219.5Notes506Acknowledgments511 Bibliography511 Exercises513 20Selfish Load Balancing517 Berthold V¨o cking20.1Introduction51720.2Pure Equilibria for Identical Machines52220.3Pure Equilibria for Uniformly Related Machines52420.4Mixed Equilibria on Identical Machines52920.5Mixed Equilibria on Uniformly Related Machines53320.6Summary and Discussion53720.7Bibliographic Notes538Bibliography540 Exercises542 21The Price of Anarchy and the Design of Scalable ResourceAllocation Mechanisms543 Ramesh Johari21.1Introduction54321.2The Proportional Allocation Mechanism54421.3A Characterization Theorem55121.4The Vickrey–Clarke–Groves Approach55921.5Chapter Summary and Further Directions564contents xi21.6Notes565Bibliography566 Exercises567IV Additional Topics22Incentives and Pricing in Communications Networks571 Asuman Ozdaglar and R.Srikant22.1Large Networks–Competitive Models57222.2Pricing and Resource Allocation–Game Theoretic Models57822.3Alternative Pricing and Incentive Approaches587Bibliography590 23Incentives in Peer-to-Peer Systems593 Moshe Babaioff,John Chuang,and Michal Feldman23.1Introduction59323.2The p2p File-Sharing Game59423.3Reputation59623.4A Barter-Based System:BitTorrent60023.5Currency60123.6Hidden Actions in p2p Systems60223.7Conclusion60823.8Bibliographic Notes608Bibliography609 Exercises610 24Cascading Behavior in Networks:Algorithmic and Economic Issues613 Jon Kleinberg24.1Introduction61324.2A First Model:Networked Coordination Games61424.3More General Models of Social Contagion61824.4Finding Influential Sets of Nodes62224.5Empirical Studies of Cascades in Online Data62724.6Notes and Further Reading630Bibliography631 Exercises632 25Incentives and Information Security633 Ross Anderson,Tyler Moore,Shishir Nagaraja,and Andy Ozment25.1Introduction63325.2Misaligned Incentives63425.3Informational Asymmetries63625.4The Economics of Censorship Resistance64025.5Complex Networks and Topology64325.6Conclusion64625.7Notes647Bibliography648xii contents26Computational Aspects of Prediction Markets651 David M.Pennock and Rahul Sami26.1Introduction:What Is a Prediction Market?65126.2Background65226.3Combinatorial Prediction Markets65726.4Automated Market Makers66226.5Distributed Computation through Markets66526.6Open Questions67026.7Bibliographic Notes671Acknowledgments672 Bibliography672 Exercises674 27Manipulation-Resistant Reputation Systems677 Eric Friedman,Paul Resnick,and Rahul Sami27.1Introduction:Why Are Reputation Systems Important?67727.2The Effect of Reputations68027.3Whitewashing68227.4Eliciting Effort and Honest Feedback68327.5Reputations Based on Transitive Trust68927.6Conclusion and Extensions69327.7Bibliographic Notes694Bibliography695 Exercises696 28Sponsored Search Auctions699 S´e bastien Lahaie,David M.Pennock,Amin Saberi,and Rakesh V.Vohra28.1Introduction69928.2Existing Models and Mechanisms70128.3A Static Model70228.4Dynamic Aspects70728.5Open Questions71128.6Bibliographic Notes712Bibliography713 Exercises715 29Computational Evolutionary Game Theory717 Siddharth Suri29.1Evolutionary Game Theory71729.2The Computational Complexity of Evolutionarily Stable Strategies72029.3Evolutionary Dynamics Applied to Selfish Routing72329.4Evolutionary Game Theory over Graphs72829.5Future Work73329.6Notes733Acknowledgments734 Bibliography734 Exercises735 Index737ForewordAs the Second World War was coming to its end,John von Neumann,arguably the foremost mathematician of that time,was busy initiating two intellectual currents that would shape the rest of the twentieth century:game theory and algorithms.In1944(16 years after the minmax theorem)he published,with Oscar Morgenstern,his Games and Economic Behavior,thus founding not only game theory but also utility theory and microeconomics.Two years later he wrote his draft report on the EDV AC,inaugurating the era of the digital computer and its software and its algorithms.V on Neumann wrote in1952thefirst paper in which a polynomial algorithm was hailed as a meaningful advance.And,he was the recipient,shortly before his early death four years later,of G¨o del’s letter in which the P vs.NP question wasfirst discussed.Could von Neumann have anticipated that his twin creations would converge half a century later?He was certainly far ahead of his contemporaries in his conception of computation as something dynamic,ubiquitous,and enmeshed in society,almost organic–witness his self-reproducing automata,his fault-tolerant network design,and his prediction that computing technology will advance in lock-step with the economy (for which he had already postulated exponential growth in his1937Vienna Colloquium paper).But I doubt that von Neumann could have dreamed anything close to the Internet, the ubiquitous and quintessentially organic computational artifact that emerged after the end of the Cold War(a war,incidentally,of which von Neumann was an early soldier and possible casualty,and that was,fortunately,fought mostly with game theory and decided by technological superiority–essentially by algorithms–instead of the thermonuclear devices that were von Neumann’s parting gift to humanity).The Internet turned the tables on students of both markets and computation.It transformed,informed,and accelerated markets,while creating new and theretofore unimaginable kinds of markets–in addition to being itself,in important ways,a market. Algorithms became the natural environment and default platform of strategic decision making.On the other hand,the Internet was thefirst computational artifact that was not created by a single entity(engineer,design team,or company),but emerged from the strategic interaction of puter scientists were for thefirst time faced with an object that they had to feel with the same bewildered awe with which economists havexiiixiv forewordalways approached the market.And,quite predictably,they turned to game theory for inspiration–in the words of Scott Shenker,a pioneer of this way of thinking who has contributed to this volume,“the Internet is an equilibrium,we just have to identify the game.”A fascinating fusion of ideas from bothfields–game theory and algorithms–came into being and was used productively in the effort to illuminate the mysteries of the Internet.It has come to be called algorithmic game theory.The chapters of this book,a snapshot of algorithmic game theory at the approximate age of ten written by a galaxy of its leading researchers,succeed brilliantly,I think,in capturing thefield’s excitement,breadth,accomplishment,and promise.Thefirst few chapters recount the ways in which the newfield has come to grips with perhaps the most fundamental cultural incongruity between algorithms and game theory:the latter predicts the agents’equilibrium behavior typically with no regard to the ways in which such a state will be reached–a consideration that would be a computer scientist’s foremost concern.Hence,algorithms for computing equilibria(Nash and correlated equilibria in games,price equilibria for markets)have been one of algorithmic game theory’s earliest research goals.This body of work has become a valuable contribu-tion to the debate in economics about the validity of behavior predictions:Efficient computability has emerged as a very desirable feature of such predictions,while com-putational intractability sheds a shadow of implausibility on a proposed equilibrium putational models that reflect the realities of the market and the Internet better than the von Neumann machine are of course at a premium–there are chapters in this book on learning algorithms as well as on distributed algorithmic mechanism design.The algorithmic nature of mechanism design is even more immediate:This elegant and well-developed subarea of game theory deals with the design of games,with players who have unknown and private utilities,such that at the equilibrium of the designed game the designer’s goals are attained independently of the agents’utilities(auctions are an important example here).This is obviously a computational problem,and in fact some of the classical results in this area had been subtly algorithmic,albeit with little regard to complexity considerations.Explicitly algorithmic work on mechanism design has,in recent years,transformed thefield,especially in the case of auctions and cost sharing(for example,how to recover the cost of an Internet service from customers who value the service by amounts known only to them)and has become the arena of especially intense and productive cross-fertilization between game theory and algorithms;these problems and accomplishments are recounted in the book’s second part.The third part of the book is dedicated to a line of investigation that has come to be called“the price of anarchy.”Selfish rational agents reach an equilibrium.The question arises:exactly how inefficient is this equilibrium in comparison to an idealized situation in which the agents would strive to collaborate selflessly with the common goal of minimizing total cost?The ratio of these quantities(the cost of an equilibrium over the optimum cost)has been estimated successfully in various Internet-related setups,and it is often found that“anarchy”is not nearly as expensive as one might have feared.For example,in one celebrated case related to routing with linear delays and explained in the“routing games”chapter,the overhead of anarchy is at most33%over the optimum solution–in the context of the Internet such a ratio is rather insignificantforeword xv and quickly absorbed by its rapid growth.Viewed in the context of the historical development of research in algorithms,this line of investigation could be called“the third compromise.”The realization that optimization problems are intractable led us to approximation algorithms;the unavailability of information about the future,or the lack of coordination between distributed decision makers,brought us online algorithms;the price of anarchy is the result of one further obstacle:now the distributed decision makers have different objective functions.Incidentally,it is rather surprising that economists had not studied this aspect of strategic behavior before the advent of the Internet.One explanation may be that,for economists,the ideal optimum was never an available option;in contrast,computer scientists are still looking back with nostalgia to the good old days when artifacts and processes could be optimized exactly.Finally,the chapters on“additional topics”that conclude the book(e.g.,on peer-to-peer systems and information markets)amply demonstrate the young area’s impressive breadth, reach,diversity,and scope.Books–a glorious human tradition apparently spared by the advent of the Internet–have a way of marking and focusing afield,of accelerating its development.Seven years after the publication of The Theory of Games,Nash was proving his theorem on the existence of equilibria;only time will tell how this volume will sway the path of algorithmic game theory.Paris,February2007Christos H.PapadimitriouPrefaceThis book covers an area that straddles twofields,algorithms and game theory,and has applications in several others,including networking and artificial intelligence.Its text is pitched at a beginning graduate student in computer science–we hope that this makes the book accessible to readers across a wide range of areas.We started this project with the belief that the time was ripe for a book that clearly develops some of the central ideas and results of algorithmic game theory–a book that can be used as a textbook for the variety of courses that were already being offered at many universities.We felt that the only way to produce a book of such breadth in a reasonable amount of time was to invite many experts from this area to contribute chapters to a comprehensive volume on the topic.This book is partitioned into four parts:thefirst three parts are devoted to core areas, while the fourth covers a range of topics mostly focusing on applications.Chapter1 serves as a preliminary chapter and it introduces basic game-theoretic definitions that are used throughout the book.Thefirst chapters of Parts II and III provide introductions and preliminaries for the respective parts.The other chapters are largely independent of one another.The authors were requested to focus on a few results highlighting the main issues and techniques,rather than provide comprehensive surveys.Most of the chapters conclude with exercises suitable for classroom use and also identify promising directions for further research.We hope these features give the book the feel of a textbook and make it suitable for a wide range of courses.You can view the entire book online at/us/9780521872829username:agt1userpassword:camb2agtMany people’s efforts went into producing this book within a year and a half of itsfirst conception.First and foremost,we thank the authors for their dedi-cation and timeliness in writing their own chapters and for providing importantxviixviii prefacefeedback on preliminary drafts of other chapters.Thanks to Christos Papadimitriou for his inspiring Foreword.We gratefully acknowledge the efforts of outside review-ers:Elliot Anshelevich,Nikhil Devanur,Matthew Jackson,Vahab Mirrokni,Herve Moulin,Neil Olver,Adrian Vetta,and several anonymous referees.Thanks to Cindy Robinson for her invaluable help with correcting the galley proofs.Finally,a big thanks to Lauren Cowles for her stellar advice throughout the production of this volume.Noam NisanTim Roughgarden´Eva TardosVijay V.VaziraniContributorsRoss AndersonComputer Laboratory University of CambridgeMoshe BabaioffSchool of Information University of California,Berkeley Avrim BlumDepartment of Computer Science Carnegie Mellon UniversityLiad BlumrosenMicrosoft ResearchSilicon ValleyJohn ChuangSchool of Information University of California,Berkeley Bruno CodenottiIstituto di Informatica e Telematica,Consiglio Nazionale delle Ricerche Yevgeniy DodisDepartment of Computer Science Courant Institute of Mathematical Sciences,New York UniversityJoan FeigenbaumComputer Science DepartmentYale UniversityMichal FeldmanSchool of Business Administrationand the Center for the Study of Rationality Hebrew University of JerusalemEric FriedmanSchool of Operations Researchand Information EngineeringCornell UniversityJason D.HartlineMicrosoft ResearchSilicon ValleyKamal JainMicrosoft ResearchRedmondRamesh JohariDepartment of Management Scienceand EngineeringStanford UniversityAnna R.KarlinDepartment of Computer Scienceand EngineeringUniversity of Washingtonxixxx contributorsMichael KearnsDepartment of Computerand Information Science University of PennsylvaniaJon KleinbergDepartment of Computer Science Cornell UniversityS´e bastien LahaieSchool of Engineeringand Applied SciencesHarvard UniversityRon LaviFaculty of Industrial Engineering and Management,The Technion Israel Institute of Technology Mohammad MahdianYahoo!ResearchSilicon ValleyYishay MansourSchool of Computer ScienceTel Aviv UniversityTyler MooreComputer Laboratory University of Cambridge Shishir NagarajaComputer Laboratory University of CambridgeNoam NisanSchool of Computer Science and EngineeringHebrew University of Jerusalem Asuman Ozdaglar Department of Electrical Engineering and Computer Science,MITAndy OzmentComputer Laboratory University of Cambridge Christos H.Papadimitriou Computer Science Division University of California,Berkeley David C.ParkesSchool of Engineeringand Applied SciencesHarvard UniversityDavid M.PennockYahoo!ResearchNew YorkTal RabinT.J.Watson Research CenterIBMPaul ResnickSchool of InformationUniversity of MichiganTim RoughgardenDepartment of Computer Science Stanford UniversityAmin SaberiDepartment of Management Science and EngineeringStanford UniversityRahul SamiSchool of InformationUniversity of MichiganMichael SchapiraSchool of Computer Scienceand EngineeringThe Hebrew University of Jerusalem James SchummerM.E.D.S.Kellogg School of Management Northwestern Universitycontributors xxiScott ShenkerEECS DepartmentUniversity of California,BerkeleyR.SrikantDepartment of Electrical and Computer Engineering and Coordinated Science Laboratory,University of Illinois at Urbana-ChampaignSiddharth SuriDepartment of Computer Science Cornell University´Eva TardosDepartment of Computer Science Cornell UniversityKasturi VaradarajanDepartment of Computer Science University of Iowa Vijay V.VaziraniCollege of ComputingGeorgia Institute of TechnologyBerthold V¨o cking Department of Computer Science RWTH Aachen UniversityRakesh V.VohraM.E.D.S.Kellogg School of Management Northwestern University Bernhard von Stengel Department of Mathematics London School of EconomicsTom WexlerDepartment of Computer Science Cornell University。

I. Introduction In this paper, we present a method for segmenting images modeled as N -ary Markov random fields (MRFs). Such image representations have proved useful for segmentation because they can explicitly model important features of actual images, such as the presence of homogeneous regions separated by sharp disconsed segmentation methods are often computationally intensive and therefore difficult to apply in a three-dimensional (3D) context. In addition, specification of the MRF parameter values is often difficult to perform. This can be done in a heuristic manner, but such an approach is strongly application-dependent and becomes very burdensome for complex models (i.e., large neighborhoods and large number of levels). Deriving unsupervised methods in which the MRF parameters are estimated from the observed data is more satisfactory, but such a task generally requires approximations in order to be mathematically feasible [1], and the corresponding amount of computations is generally much higher than for a segmentation operation. In order to overcome such difficulties, Devijver and Dekesel [2] proposed an unsupervised segmentation approach based on a hidden markov model (HMM) that belongs to a special class of unilateral MRFs: Pickard random fields (PRFs). The PRF is observed through an independent Gaussian process and the labels as well as the model parameters are estimated using maximum likelihood techniques. Because of the specific properties of PRF models, a significant reduction of the computational burden is achieved, and application of such methods to 3D problems can be envisioned. However, three kinds of difficulties remain: firstly, from a theoretical standpoint, the estimated MRF parameters are not necessarily consistent with the assumed stationarity of the model; secondly, the likelihood function of the observation parameters presents attractive singular points, as it is well-known in Gaussian mixture identification problems [3], and this hinders the convergence of the estimation procedure; thirdly, the convergence of the estimation procedure is made even more difficult by the fairly large number of parameters that need to be estimated.