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On the Trade-Offs Between Modeling Power and Algorithmic Complexity On the Trade-offs between Modeling Power and Algorithmic Complexity Chun Ye Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2016 c 2016 Chun Ye All Rights Reserved ABSTRACT On the Trade-offs between Modeling Power and Algorithmic Complexity Chun Ye Mathematical modeling is a central component of operations research. Most of the academic research in our field focuses on developing algorithmic tools for solving various mathematical problems arising from our models. However, our procedure for selecting the best model to use in any particular application is ad hoc. This dissertation seeks to rigorously quantify the trade-offs between various design criteria in model construction through a series of case studies. The hope is that a better understanding of the pros and cons of different models (for the same application) can guide and improve the model selection process. In this dissertation, we focus on two broad types of trade-offs. The first type arises naturally in mechanism or market design, a discipline that focuses on developing optimization models for complex multi-agent systems. Such systems may require satisfying multiple objectives that are potentially in conflict with one another. Hence, finding a solution that simultaneously satisfies several design requirements is challenging. The second type addresses the dynamics between model complexity and computational tractability in the context of approximation algorithms for some discrete optimization problems. The need to study this type of trade-offs is motivated by certain industry problems where the goal is to obtain the best solution within a reasonable time frame. Hence, being able to quantify and compare the degree of sub-optimality of the solution obtained under different models is helpful. Chapters 2-5 of the dissertation focus on trade-offs of the first type and Chapters 6-7 the second type. Table of Contents List of Figures vi List of Tables vii 1 Introduction 1 1.1 Trade-offs in Multi-Agent Optimization Models . .2 1.2 Model Predictive Power versus Computational Tractability Trade-offs . .8 2 A Note on the Assignment Problem with Uniform Preferences 11 2.1 Introduction . 11 2.1.1 Related Literature . 13 2.2 Preliminaries . 14 2.2.1 Model and Definitions . 14 2.2.2 Desirable Properties . 15 2.2.3 The Extended Probabilistic Serial Mechanism . 17 2.3 Main Results . 18 2.3.1 Non-uniqueness of Ordinally Efficient and Envy Free Assignments . 18 2.3.2 Impossibility Results . 19 3 Cake Cutting Algorithms for Piecewise Constant and Piecewise Uniform Valua- tions 25 3.1 Introduction . 25 3.1.1 Related Work . 29 i 3.2 Preliminaries . 31 3.2.1 Model . 31 3.2.2 Properties of Allocations . 32 3.2.3 Properties of Cake Cutting Algorithms . 33 3.2.4 Relationship between the Properties . 34 3.2.5 The Free Disposal Assumption . 34 3.2.6 The (Random) Assignment Problem and its Relationship to Cake Cutting . 35 3.3 CCEA | Controlled Cake Eating Algorithm . 37 3.3.1 CCEA for Piecewise Uniform Valuations . 40 3.4 MEA | Market Equilibrium Algorithm . 41 3.4.1 MEA for Piecewise Uniform Valuations . 44 3.5 MCSD | Mixed Constrained Serial Dictatorship Algorithm . 46 3.6 Discussion . 51 4 A Generalization of the Probabilistic Serial Mechanism and its Relationship to the Leximin Allocation 55 4.1 Introduction . 55 4.1.1 Our Contributions . 57 4.1.2 Related Work . 57 4.1.3 Model and Notation . 59 4.1.4 The Generalized Leximin Allocation Vector . 60 4.2 Computing a Generalized Leximin Allocation . 61 4.3 The Equivalence between the Generalized Probablistic Serial (GPS) Mechanism and the Generalized Leximin Mechanism . 63 4.3.1 Dichotomous Preference Domain . 65 4.3.2 Full Preference Domain . 68 4.4 Properties of the Generalized Leximin Mechanism . 71 4.5 Discussion . 75 5 Approximately Optimal Mechanisms for Strategyproof Facility Location: Minimiz- ing Lp Norm of Costs 76 ii 5.1 Introduction . 76 5.2 Model . 80 5.3 The Median Mechanism . 82 5.4 Randomized Mechanisms . 86 5.4.1 Mixing Dictatorships and Generalized Medians with the Optimal Location . 86 5.4.2 Optimality of the LRM Mechanism for 2 Agents . 89 5.5 Discussion . 94 6 Approximation Algorithms for the Incremental Knapsack Problem 95 6.1 Introduction . 95 6.1.1 Scheduling Interpretation . 97 6.1.2 Related Work . 98 6.1.3 Our Contributions . 99 6.2 Hardness Results for ................................... 101 DIK 6.3 A Constant Factor Approximation Algorithm . 104 6.3.1 Improving the Approximation Guarantee . 106 6.4 A PTAS for ....................................... 107 DIIK 6.4.1 Discussion . 111 6.5 Continuous Knapsack with Linear Capacity . 111 6.5.1 Incremental Subset Sum with Linear Capacity . 115 6.5.2 Discussion . 117 6.6 Piecewise Linear Capacity function . 117 6.6.1 Discussion . 118 7 Capacity Constrained Assortment Optimization under the Markov Chain based Choice Model 119 7.1 Introduction . 119 7.1.1 Our Contributions . 121 7.1.2 The Markov Chain Model and Notations . 123 7.1.3 Outline . 124 7.2 Hardness of Approximation . 124 iii 7.2.1 APX-hardness for Cardinality Constraint with Uniform Prices . 124 7.2.2 Totally-Unimodular Constraints . 125 7.3 Special Case: Uniform Price Items . 126 7.3.1 Constant Factor Approximation Algorithm . 126 7.3.2 Bad Examples for Arbitrary Prices . 128 7.4 Local Ratio based Algorithm Design . 131 7.4.1 High-Level Ideas . 131 7.4.2 Instance Update in Local Ratio Algorithm . 132 7.4.3 Structural Properties of the Updates . 134 7.4.4 Warm-up: Exact algorithm for the Unconstrained Problem . 135 7.5 Cardinality Constrained Assortment Optimization for General Case . 137 7.5.1 Technical Lemmas . 138 7.5.2 Analysis of the Local-Ratio Algorithm . 141 7.6 Capacity Constrained Assortment Optimization for General Case . 143 7.6.1 Analysis . 144 7.7 Computational Experiments . 147 7.7.1 A Mixed-Integer Programming Formulation . 148 7.7.2 Settings Tested . 148 7.7.3 Results . 149 7.8 Discussion . 151 I Bibliography 154 Bibliography 155 II Appendices 167 A Cake Cutting Algorithms for Piecewise Constant and Piecewise Uniform Valua- tions 168 iv B Approximately Optimal Mechanisms for Strategyproof Facility Location: Minimiz- ing Lp Norm of Costs 177 B.1 An Alternative Definition of Individual Cost . 177 B.2 Omitted Proofs from Section 4.2 . 178 B.3 Alternative Assumptions in Section 4.2 . 183 C Approximation Algorithms for the Incremental Knapsack Problem 185 C.1 Proof of Proposition 6.2 . 185 C.2 Proof of Lemma C.2 . 186 C.3 Proof of Proposition 6.5 . 189 C.4 Proof of Proposition 6.4 . ..
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