More than the sum of its parts : compact preference representation over combinatorial domains - Contents

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More than the sum of its parts : compact preference representation over

combinatorial domains

Uckelman, J.D.

Publication date 2009

Link to publication

Citation for published version (APA):

Uckelman, J. D. (2009). More than the sum of its parts : compact preference representation over combinatorial domains. Institute for Logic, Language and Computation.

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Contents

Acknowledgments xi 1 Introduction 1 2 Languages 9 2.1 Introduction . . . 9 2.2 Notation . . . 9 2.2.1 Propositional Logic . . . 9

2.2.2 Utility Functions, Goalbases, and Languages . . . 11

2.3 Related Languages . . . 14

2.3.1 CP-Nets . . . 14

2.3.2 Penalty Logic . . . 16

2.3.3 Weighted and Distance-Based Logics for Cardinal Disutility 18 2.3.4 Propositional Languages for Ordinal Preferences . . . 19

2.3.5 Weighted Description Logics . . . 23

2.3.6 Boolean Games . . . 23

2.3.7 Valued Constraint Satisfaction Problems . . . 24

2.3.8 Generalized Additive Independence . . . 25

2.3.9 Coalitional Games . . . 26 2.3.10 Bidding Languages . . . 26

I

Theory

29

3 Expressivity 31 3.1 Introduction . . . 31 3.2 Preliminaries . . . 32 3.3 Related Work . . . 34

3.4 Expressivity of Sum Languages . . . 35 vii

3.4.1 Goalbase Equivalences . . . 35

3.4.2 Uniqueness . . . 36

3.4.3 Correspondences . . . 43

3.4.4 Summary . . . 49

3.5 Expressivity of Max Languages . . . 49

3.5.1 Superfluous Goals . . . 51

3.5.2 Goalbase Equivalences . . . 53

3.5.3 Correspondences . . . 55

3.5.4 Summary . . . 56

3.6 Odds and Ends . . . 58

3.7 Conclusion . . . 59

4 Succinctness 61 4.1 Introduction . . . 61

4.2 Preliminaries . . . 61

4.3 Related Work . . . 65

4.4 Succinctness of Sum Languages . . . 68

4.4.1 Some Basic Succinctness and Equivalence Results . . . 68

4.4.2 Equivalence via Goalbase Translation . . . 70

4.4.3 Strict Succinctness and Incomparability, by Counterexample 71 4.4.4 Strict Succinctness, Nonconstructively . . . 74

4.4.5 Summary . . . 79

4.5 Succinctness of Max Languages . . . 79

4.5.1 Absolute Succinctness . . . 79 4.5.2 Relative Succinctness . . . 86 4.5.3 Summary . . . 88 4.6 Cross-Aggregator Succinctness . . . 90 4.7 Conclusion . . . 93 5 Complexity 95 5.1 Introduction . . . 95 5.2 Background . . . 95

5.3 _{The Decision Problems max-util, min-util, and max-cuf . . . 100}

5.4 Related Work . . . 101

5.5 _{The Complexity of max-util and min-util . . . 103}

5.5.1 _{Hardness Results for max-util . . . 104}

5.5.2 _{Easiness Results for max-util . . . 109}

5.5.3 _{The Complexity of min-util . . . 111}

5.5.4 Summary . . . 115

5.6 The Complexity of Collective Utility Maximization . . . 116

5.6.1 Summary . . . 121

5.7 _{An Alternate Formulation of max-util . . . 122}

5.7.1 _{Revising the max-util Decision Problem . . . 122} viii

5.7.2 _{Horn Clauses, Logic Programming, and hornsat . . . 125}

5.7.3 Finding P-Complete Goalbase Languages . . . 126

5.7.4 Discussion . . . 129 5.8 Conclusion . . . 130

II

Applications

133

6 Combinatorial Auctions 135 6.1 Introduction . . . 135 6.2 Auctions . . . 135 6.3 Bidding Languages . . . 138

6.3.1 The XOR, OR, and OR∗ Languages . . . 139

6.3.2 Goalbase Bidding Languages . . . 139

6.3.3 Succinctness . . . 140

6.4 Winner Determination . . . 145

6.4.1 Notation . . . 145

6.4.2 The Winner Determination Problem . . . 146

6.4.3 An IP Formulation of the WDP . . . 147

6.4.4 Branch-and-Bound WDP Algorithms . . . 150

6.5 Heuristics for Winner Determination . . . 154

6.5.1 Expansion and Branching Policies . . . 154

6.5.2 Heuristics for Positive Cubes . . . 155

6.5.3 Heuristics for Positive Clauses . . . 158

6.5.4 Heuristics for Cubes . . . 159

6.6 Experimental Setup . . . 161

6.6.1 Principles for Generating Realistic Data . . . 161

6.6.2 Data Generation . . . 163

6.7 Experimental Results . . . 166

6.7.1 First Solver . . . 166

6.7.2 Second Solver and CPLEX . . . 171

6.7.3 Comparison of Solvers . . . 175 6.8 Conclusion . . . 179 7 Voting 181 7.1 Introduction . . . 181 7.2 Background . . . 181 7.3 Multi-Winner Elections . . . 185

7.3.1 Some Methods for Committee Election . . . 185

7.3.2 Similar Committees Need Not Be Similarly Preferable . . . 188

7.4 Simulating Voting Methods Using Goalbases . . . 191

7.5 The Complexity of Deciding Winning Slates . . . 194

7.6 Extending Single-Winner Voting Methods . . . 196 ix

7.7 Future Work . . . 199 7.8 Conclusion . . . 201 8 Conclusion 203 Bibliography 206 List of Symbols 222 Index 222 Samenvatting 223 Abstract 225 x