Algorithmics of matching under preferences
Matching problems with preferences are all around us: they arise when agents seek to be allocated to one another on the basis of ranked preferences over potential outcomes. Efficient algorithms are needed for producing matchings that optimise the satisfaction of the agents according to their prefere...
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| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
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Singapore
World Scientific Publishing Co. Pte. Ltd
2013
World Scientific Publishing Company WORLD SCIENTIFIC WSPC |
| Vydání: | 1 |
| Edice: | Series on theoretical computer science |
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| ISBN: | 9789814425247, 9814425249 |
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| Abstract | Matching problems with preferences are all around us: they arise when agents seek to be allocated to one another on the basis of ranked preferences over potential outcomes. Efficient algorithms are needed for producing matchings that optimise the satisfaction of the agents according to their preference lists. In recent years there has been a sharp increase in the study of algorithmic aspects of matching problems with preferences, partly reflecting the growing number of applications of these problems worldwide. The importance of the research area was recognised in 2012 through the award of the Nobel Prize in Economic Sciences to Alvin Roth and Lloyd Shapley. |
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| AbstractList | Matching problems with preferences are all around us: they arise when agents seek to be allocated to one another on the basis of ranked preferences over potential outcomes. Efficient algorithms are needed for producing matchings that optimise the satisfaction of the agents according to their preference lists.In recent years there has been a sharp increase in the study of algorithmic aspects of matching problems with preferences, partly reflecting the growing number of applications of these problems worldwide. The importance of the research area was recognised in 2012 through the award of the Nobel Prize in Economic Sciences to Alvin Roth and Lloyd Shapley.This book describes the most important results in this area, providing a timely update to The Stable Marriage Problem: Structure and Algorithms (D Gusfield and R W Irving, MIT Press, 1989) in connection with stable matching problems, whilst also broadening the scope to include matching problems with preferences under a range of alternative optimality criteria.Sample Chapter(s)Foreword (46 KB)Chapter 1: Preliminary Definitions, Results and Motivation (430 KB)Contents:Preliminary Definitions, Results and MotivationStable Matching Problems:The Stable Marriage Problem: An UpdateSM and HR with IndifferenceThe Stable Roommates ProblemFurther Stable Matching ProblemsOther Optimal Matching Problems:Pareto Optimal MatchingsPopular MatchingsProfile-Based Optimal MatchingsReadership: Students and Professionals interested in algorithms, especially in the study of algorithmic aspects of matching problems with preferences. Matching problems with preferences are all around us: they arise when agents seek to be allocated to one another on the basis of ranked preferences over potential outcomes. Efficient algorithms are needed for producing matchings that optimise the satisfaction of the agents according to their preference lists. In recent years there has been a sharp increase in the study of algorithmic aspects of matching problems with preferences, partly reflecting the growing number of applications of these problems worldwide. The importance of the research area was recognised in 2012 through the award of the Nobel Prize in Economic Sciences to Alvin Roth and Lloyd Shapley. Matching problems with preferences are all around us: they arise when agents seek to be allocated to one another on the basis of ranked preferences over potential outcomes. Efficient algorithms are needed for producing matchings that optimise the satisfaction of the agents according to their preference lists. In recent years there has been a sharp increase in the study of algorithmic aspects of matching problems with preferences, partly reflecting the growing number of applications of these problems worldwide. The importance of the research area was recognised in 2012 through the award of the Nobel Prize in Economic Sciences to Alvin Roth and Lloyd Shapley. This book describes the most important results in this area, providing a timely update to The Stable Marriage Problem: Structure and Algorithms (D Gusfield and R W Irving, MIT Press, 1989) in connection with stable matching problems, whilst also broadening the scope to include matching problems with preferences under a range of alternative optimality criteria. |
| Author | Manlove, David F |
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| Keywords | House Allocation Problem Algorithms Stable Marriage Problem Stable Roomates Problem Hospitals / Residents Problem Preferences Matching Problems |
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| Notes | Includes bibliographical references (p. 417-460) and index |
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| Publisher | World Scientific Publishing Co. Pte. Ltd World Scientific Publishing Company WORLD SCIENTIFIC WSPC |
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| Snippet | Matching problems with preferences are all around us: they arise when agents seek to be allocated to one another on the basis of ranked preferences over... |
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| SubjectTerms | Computer algorithms Econometrics Formal Specification (Software Engineering, Mathematical Logic) Game Theory Matching theory Mathematical Economics |
| SubjectTermsDisplay | Matching theory |
| TableOfContents | Algorithmics of matching under preferences -- Preface -- Foreword -- Acknowledgments -- List of Figures -- List of Tables -- List of Algorithms -- Chapter 1: Preliminary Definitions, Results and Motivation -- Part 1: Stable Matching Problems -- Chapter 2: The Stable Marriage problem: An update -- Chapter 3: The Stable Marriage and Hospitals/Residents Problems with Indifference -- Chapter 4: The Stable Roommates Problem -- Chapter 5: Further Stable Matching Problems -- Part 2: Other Optimal Matching Problems -- Chapter 6: Pareto Optimal Matchings -- Chapter 7: Popular Matchings -- Chapter 8: Profile-Based Optimal Matchings -- Bibliography -- Glossary of Symbols -- Index. 1.5.4 Maximum utility matchings -- 1.5.5 Popular matchings -- 1.5.6 Profile-based optimal matchings -- 1.5.7 Extensions of ha -- 1.5.8 Motivation -- Stable Matching Problems -- 2. The Stable Marriage problem: An update -- 2.1 Introduction -- 2.2 The 12 open problems of Gusfield and Irving -- 2.2.1 Introduction -- 2.2.2 1. Maximum number of stable matchings -- 2.2.3 2. The "divorce digraph" -- 2.2.4 3. Parallel algorithms for stable marriage -- 2.2.5 4. Batch stability testing -- 2.2.6 5. Structure of stable marriage with ties -- 2.2.7 6. Sex-equal matching -- 2.2.8 7. Lying and egalitarian matchings -- 2.2.9 10. Succinct certificates -- 2.2.10 11. Algorithmic improvements -- 2.3 The Subramanian and Feder papers -- 2.3.1 Subramanian: sri and network stability -- 2.3.2 Feder: sri and 2-sat -- 2.3.3 Other fixed-point approaches -- 2.4 Linear programming approaches -- 2.5 Constraint programming approaches -- 2.5.1 Introduction -- 2.5.2 Preliminaries -- 2.5.3 Overview of the csp model -- 2.5.4 Arc consistency in the csp model -- 2.6 Paths to stability -- 2.6.1 Introduction -- 2.6.2 The Roth-Vande Vate Mechanism -- 2.6.3 The Random Order Mechanism -- 2.6.4 Other decentralised algorithms -- 2.7 Median stable matchings -- 2.8 Size versus stability -- 2.9 Strategic issues -- 2.10 Further results -- 2.10.1 Stable Marriage problem with Forbidden pairs -- 2.10.2 Balanced stable matchings -- 2.10.3 Rationalizing matchings -- 2.10.4 Dinitz conjecture and stable marriage theory -- 2.10.5 The marriage graph -- 2.10.6 Sampling and counting -- 2.10.7 Online algorithms -- 2.10.8 Finding "good" stable matchings: unified approach -- 2.10.9 Locally stable matchings -- 2.10.10 Miscellaneous results -- 2.11 Conclusions and open problems -- 3. SM and HR with indifference -- 3.1 Introduction -- 3.2 Weak stability -- 3.2.1 Existence of a weakly stable matching 7.2.6 Popular matchings in hat 5.6 3D stable matching problems -- 5.6.1 3D variants of sm -- 5.6.1.1 Strictly-ordered preferences over pairs -- 5.6.1.2 Preferences over pairs with ties -- 5.6.1.3 Lexicographic preferences over pairs of agents -- 5.6.1.4 Preferences over individual agents -- 5.6.2 3D variants of sr -- 5.6.2.1 Strictly-ordered preferences over pairs -- 5.6.2.2 Preference lists over pairs with ties -- 5.6.2.3 Preferences over individual agents -- 5.6.2.4 Three-Way Kidney Transplant -- 5.6.2.5 Geometric 3dsr -- 5.7 Exchange-stable matching problems -- 5.7.1 Exchange-stability as a solution concept -- 5.7.2 Exchange-blocking coalitions -- 5.7.3 Stable matchings that are exchange-stable -- 5.8 Two additional stable matching problems -- 5.8.1 Bistable matching problems -- 5.8.2 The Cycle Stable Roommates problem -- 5.9 Conclusions and open problems -- Other Optimal Matching Problems -- 6. Pareto optimal matchings -- 6.1 Introduction -- 6.2 House Allocation problem -- 6.2.1 Strictly-ordered preferences -- 6.2.1.1 Testing for Pareto optimality -- 6.2.1.2 Maximum Pareto optimal matchings -- 6.2.1.3 Other results for Pareto optimal matchings -- 6.2.1.4 Matchings in the cor -- 6.2.2 Preference lists with ties -- 6.2.2.1 Characterisation of Pareto optimal matchings -- 6.2.2.2 Matchings in the core -- 6.3 Capacitated House Allocation problem -- 6.4 Hospitals / Residents problem -- 6.5 Stable Roommates problem -- 6.5.1 Introduction -- 6.5.2 Preliminary observations -- 6.5.3 Characterising Pareto optimal matchings -- 6.5.4 Maximum Pareto optimal matchings -- 6.5.5 Coalition formation games -- 6.6 Conclusions and open problems -- 7. Popular matchings -- 7.1 Introduction -- 7.2 House Allocation problem -- 7.2.1 Introduction -- 7.2.2 Characterising popular matchings -- 7.2.3 Finding a popular matching -- 7.2.4 Maximum popular matchings -- 7.2.5 Structure of popular matchings 4.5.4 Super-stable matchings -- 4.6 "Almost stable" matchings -- 4.6.1 Introduction -- 4.6.2 Hardness results -- 4.6.3 Matchings with a constant no. of blocking pairs -- 4.6.4 Bounded length preference lists -- 4.6.5 Open problems -- 4.7 Globally-ranked pairs -- 4.7.1 Definitions and motivation for the srti-grp model -- 4.7.2 Globally acyclic preferences -- 4.7.3 Weakly / strongly stable matchings in srti-grp -- 4.7.4 Related work -- 4.8 Other extensions of sr -- 4.8.1 Introduction -- 4.8.2 Stable Roommates problem with Forbidden Pairs -- 4.8.3 Stable Crews problem -- 4.8.4 Stable Fixtures problem -- 4.8.5 Stable Multiple Activities problem -- 4.8.6 Stable Allocation problem -- 4.8.7 Stable Roommates problem with Choice Functions -- 4.8.8 Coalition Formation Games -- 4.9 Conclusions and open problems -- 5. Further stable matching problems -- 5.1 Introduction -- 5.2 hr with lower and common quotas -- 5.2.1 Introduction -- 5.2.2 hr with lower quotas (model 1) -- 5.2.3 hr with lower quotas (model 2) -- 5.2.4 hr with common quotas -- 5.2.5 Classified stable matching -- 5.3 hr with couples -- 5.3.1 Introduction -- 5.3.2 Problem definition and preliminary results -- 5.3.3 Algorithmic results -- 5.3.4 Consistent couples -- 5.3.5 Inseparable couples -- 5.4 Many-many stable matching -- 5.4.1 Introduction -- 5.4.2 Definition of the basic wf model -- 5.4.3 wf-1: preferences over individuals -- 5.4.4 wf-2: group preferences -- 5.5 The Student-Project Allocation Problem -- 5.5.1 Introduction -- 5.5.2 Lecturer preferences over students: spa-s -- 5.5.2.1 Introduction -- 5.5.2.2 Overview of Algorithm spa-s-student -- 5.5.2.3 Properties of stable matchings in an instance of spa-s -- 5.5.2.4 Lecturer-oriented algorithm -- 5.5.2.5 Open problems -- 5.5.3 Lecturer preferences over projects -- 5.5.4 Lecturer preferences over student-project pairs Intro -- Contents -- Preface -- Foreword -- Acknowledgments -- List of Figures -- List of Tables -- List of Algorithms -- 1. Preliminary definitions, results and motivation -- 1.1 Introduction -- 1.1.1 Remit of this book -- 1.1.1.1 Matching under preferences -- 1.1.1.2 Free-for-all markets -- 1.1.1.3 Centralised matching schemes -- 1.1.2 The matching problems under consideration -- 1.1.2.1 Classification of matching problems -- 1.1.2.2 Bipartite matching problems with two-sided preferences -- 1.1.2.3 Bipartite matching problems with one-sided preferences -- 1.1.2.4 Non-bipartite matching problems with preferences -- 1.1.2.5 Further problem variants -- 1.1.3 Existing literature on matching problems -- 1.1.3.1 Algorithms and complexity literature -- 1.1.3.2 Game theory and economics literature -- 1.1.3.3 Algorithmic mechanism design literature -- 1.1.4 Contribution of this book -- 1.1.4.1 General overview -- 1.1.4.2 Chapter outline -- 1.1.4.3 What the book does not contribute -- 1.1.5 Outline of this chapter -- 1.2 Matchings in graphs -- 1.3 The Hospitals / Residents problem (hr) -- 1.3.1 Introduction -- 1.3.2 Key definitions -- 1.3.3 Key results (up to 1989) -- 1.3.4 Stable Marriage problem (sm) -- 1.3.4.1 Key definitions -- 1.3.4.2 Key results (up to 1989) -- 1.3.4.3 Rotations -- 1.3.5 Hospitals / Residents problem with indifference -- 1.3.6 Other variants of hr -- 1.3.6.1 Couples -- 1.3.6.2 Many-many stable matchings -- 1.3.6.3 Master lists -- 1.3.7 Motivation -- 1.4 The Stable Roommates problem (sr) -- 1.4.1 Introduction -- 1.4.2 Key definitions -- 1.4.3 Key results (up to 1989) -- 1.4.4 Rotations -- 1.4.5 Stable Roommates problem with indifference -- 1.4.6 Motivation -- 1.5 The House Allocation problem (ha) and its variants -- 1.5.1 Introduction -- 1.5.2 Formal definition of ha and hm -- 1.5.3 Pareto optimal matchings 3.2.2 Absence of a lattice structure -- 3.2.3 Sizes of weakly stable matchings -- 3.2.4 NP-hardness of max smti -- 3.2.5 Parameterized complexity of max smti -- 3.2.6 Approximability of max smti and max hrt -- 3.2.6.1 Overview of approximability results for max smti -- 3.2.6.2 Kiraly's approximation algorithm -- 3.2.6.3 Comparison of approximation algorithms for max smti -- 3.2.6.4 Heuristics for max hrt -- 3.2.6.5 "Cloning" hospitals -- 3.2.7 Other problems involving weak stability -- 3.3 Strong stability -- 3.3.1 Existence of a strongly stable matching -- 3.3.2 Rural Hospitals Theorem for hrt -- 3.3.3 Strongly stable matchings form a lattice -- 3.3.4 Finding a strongly stable matching -- 3.4 Super-stability -- 3.4.1 Existence of a super-stable matching -- 3.4.2 Rural Hospitals Theorem for hrt -- 3.4.3 Super-stable matchings form a lattice -- 3.4.4 Finding a super-stable matching -- 3.4.5 Optimal super-stable matchings -- 3.5 Other results -- 3.5.1 Semi-strong stability -- 3.5.2 Many-many strongly stable matchings -- 3.5.3 Partially-ordered preference lists -- 3.6 Conclusions and open problems -- 4. The Stable Roommates problem -- 4.1 Introduction -- 4.2 Updates to open problems 8-12 from Gusfield & -- Irving -- 4.2.1 8: Solvable Roommates Instances -- 4.2.2 9: Roommates to Marriage -- 4.2.3 10: Succinct Certificates -- 4.2.4 11: Algorithmic Improvements -- 4.2.5 12: Optimal Roommates -- 4.2.6 12.1: Linear Programming for Roommates -- 4.3 Stable partitions -- 4.3.1 Introduction -- 4.3.2 Definition and structure of stable partitions -- 4.3.3 Algorithms for finding a stable partition -- 4.3.4 Maximum stable matchings -- 4.3.5 Stable half-matchings -- 4.3.6 P-stable matchings and absorbing sets -- 4.4 Mirror posets and median graphs -- 4.5 Indifference -- 4.5.1 Introduction -- 4.5.2 Weakly stable matchings -- 4.5.3 Strongly stable matchings |
| Title | Algorithmics of matching under preferences |
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