New Approaches to Integer Programming

NEW APPROACHES TO INTEGER PROGRAMMING A Thesis Presented to The Academic Faculty by Karthekeyan Chandrasekaran In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in Algorithms, Combinatorics and Optimization in the School of Computer Science Georgia Institute of Technology August 2012 NEW APPROACHES TO INTEGER PROGRAMMING Approved by: Santosh Vempala, Advisor Santanu Dey School of Computer Science School of Industrial and Systems Georgia Institute of Technology Engineering Georgia Institute of Technology Shabbir Ahmed Richard Karp School of Industrial and Systems Department of Electrical Engineering Engineering and Computer Sciences Georgia Institute of Technology University of California, Berkeley William Cook George Nemhauser School of Industrial and Systems School of Industrial and Systems Engineering Engineering Georgia Institute of Technology Georgia Institute of Technology Date Approved: 24th May 2012 To my parents, Nalini and Chandrasekaran. iii ACKNOWLEDGEMENTS My biggest thanks to my advisor Santosh Vempala. His constant reminder to strive for excellence has helped me push myself and motivated me to be a better researcher. I can only wish to attain as much clarity in my explanations as he does. I am very grateful to the ACO program led by Robin Thomas. It has provided me an invaluable opportunity to appreciate the various aspects of theoretical research. I consider myself lucky for having been able to interact with its exceptional faculty. I also thank ARC and NSF for their generous research and travel support. Many faculty had a positive impact on me during my time at Georgia Tech. I thank Bill Cook for his alluring treatment of linear inequalities. I thank Shabbir Ahmed, Santanu Dey, George Nemhauser and Prasad Tetali for their feedback, thoughtful questions and for all the support, help and advice. I thank all the theory faculty including Nina Balcan, Dana Randall and Vijay Vazirani for their constant encouragement. I thank Richard Karp for exposing me to a refreshing simplistic take on theoretical computer science and for all the help and advice. I thank my collaborators Daniel Dadush, Amit Deshpande, Navin Goyal, Bernhard Hauepler, and Laci Vegh for all the insightful discussions. Amit and Navin have been wonderful mentors and more importantly, friends. I thank all the theory students and postdocs at Georgia Tech for maintaining an enrich- ing research environment. Subruk deserves a particular mention for being an amazing friend and a limitless source of information. I thank Elena, Laci and Will for all their help, advice and outings. I thank Akshay, Amanda, Anand, Elena, Florin, Luis, Prateek, Sarah(s), Steven, Subruk, and Ying for providing timely constructive feedback on my writings. Finally, I thank my girlfriend Sneha for being a pillar of strength and a source of joy. She has been very patient in all my endeavours and I owe more to her love than I can write here. iv TABLE OF CONTENTS DEDICATION ...................................... iii ACKNOWLEDGEMENTS .............................. iv LIST OF FIGURES .................................. viii SUMMARY ........................................ ix I INTRODUCTION ................................. 1 1.1 Contributions . .4 1.2 Tools and Directions . .7 II COMBINATORIAL APPROACH: FEEDBACK VERTEX SET IN IM- PLICIT HITTING SET MODEL ....................... 9 2.1 Implicit Hitting Set Problems . .9 2.1.1 Generic vs online algorithms . 12 2.1.2 Results . 13 2.2 Preliminaries . 14 2.2.1 Related Work . 14 2.3 Algorithms . 15 2.3.1 An Exact Algorithm . 15 2.3.2 Augment-BFS . 16 2.3.3 Hit-and-Prune . 17 2.4 FVS in Random Graphs . 17 2.4.1 Grow induced BFS . 19 2.4.2 Analysis . 20 2.4.3 Lower Bound . 31 2.5 Planted FVS . 33 2.5.1 Model . 34 2.5.2 Properties of Planted Subset . 35 2.5.3 Hitting Set is Planted FVS . 36 2.5.4 Pruning Algorithm to recover Planted FVS . 37 2.6 Conclusion . 39 v III GEOMETRIC APPROACH: DISCREPANCY-BASED ALGORITHM FOR IP ........................................ 40 3.1 Introduction . 40 3.1.1 Motivation . 41 3.1.2 Results . 42 3.2 New tool: Discrepancy for IP . 43 3.3 New model for random IP . 44 3.4 Preliminaries . 45 3.4.1 Related work . 45 3.4.2 Concentration Inequalities . 46 3.5 Discrepancy of Gaussian Matrix . 47 3.5.1 Partial Vector . 49 3.5.2 Bounding Lengths of Gaussian Subvectors . 53 3.5.3 Discrepancy Bound . 54 3.6 Phase-transition in random IP . 57 3.6.1 Radius for Integer Infeasibility . 57 3.6.2 Threshold Radius . 60 3.7 Algorithm to find Integer Point . 62 3.7.1 Algorithmic Partial Vector . 64 3.7.2 Finding an Integer Point . 66 3.8 Conclusion . 69 IV LP APPROACH: CONVERGENCE OF CUTTING PLANES FOR MIN- COST PERFECT MATCHING ........................ 71 4.1 Introduction . 71 4.1.1 Motivation . 73 4.1.2 Results . 75 4.2 Perfect Matchings and Linear Programs . 76 4.3 A Cutting Plane Algorithm for Perfect Matching . 77 4.3.1 Half-integrality for faster cut-generation . 78 4.3.2 Retain cuts with positive dual values . 79 4.3.3 Algorithm C-P-Matching . 79 vi 4.4 Factor-critical matchings . 80 4.5 Analysis . 83 4.5.1 Overview and Organization . 83 4.5.2 Contractions and Half-Integrality . 86 4.5.3 Structure of Dual Solutions . 90 4.5.4 Uniqueness . 102 4.5.5 Half-Integral Primal-Dual Algorithm . 105 4.5.6 Convergence . 112 4.6 A Half-Integral Primal-Dual Algorithm for Minimum-cost Perfect Matching 119 4.6.1 Algorithm . 119 4.6.2 Optimality . 122 4.6.3 Running Time . 123 4.7 Conclusion . 128 V CONCLUSIONS AND FUTURE DIRECTIONS ............. 130 5.1 Planted Feedback Vertex Set . 130 5.2 Hereditary Discrepancy . 131 5.3 Random IP Instances . 131 5.4 Analyzing the Cutting Plane Method . 131 REFERENCES ..................................... 132 vii LIST OF FIGURES 1 Exact Algorithm for IHS . 16 2 Algorithm Augment-BFS . 17 3 Algorithm Hit-and-Prune(k) . 17 4 BFS Exploration . 19 5 Algorithm Grow-induced-BFS . 21 6 Planted Model . 34 7 Random IP instance P (n; m; x0;R)....................... 45 8 Algorithm Round-IP . 63 9 Algorithm Edge-Walk . 65 10 Cutting plane method for matchings . 74 11 Non-integral Solution to Bipartite Relaxation . 76 12 Counterexample to the half-integrality conjecture . 78 13 Algorithm C-P-Matching . 81 14 Contraction operation: image of the dual and the new cost function . 88 15 Algorithm Positively-critical-dual-opt . 97 16 Possible primal updates in the Half-integral Primal-Dual algorithm . 106 17 The Half-Integral Primal-Dual Algorithm . 108 18 The Half-Integral Perfect Matching Algorithm . 122 viii SUMMARY Integer Programming (IP) is a powerful and widely-used formulation for combinatorial problems. The study of IP over the past several decades has led to fascinating theoretical developments, and has improved our ability to solve discrete optimization problems arising in practice. This thesis makes progress on algorithmic solutions for IP by building on combinatorial, geometric and Linear Programming (LP) approaches. We use a combinatorial approach to give an approximation algorithm for the Feedback Vertex Set problem in a recently developed Implicit Hitting Set framework [46, 47]. Our algorithm is a simple online algorithm which finds a nearly optimal Feedback Vertex Set in random graphs. We also propose a planted model for Feedback Vertex Set and show that an optimal hitting set for a polynomial number of subsets is sufficient to recover the planted Feedback Vertex Set. Next, we present an unexplored geometric connection between integer feasibility and the classical notion of discrepancy of matrices [73]. We exploit this connection to show a phase transition from infeasibility to feasibility in random IP instances. A recent algorithm for small discrepancy solutions [59] leads to an efficient algorithm to find an integer point for random IP instances that are feasible with high probability. Finally, we give a provably efficient cutting-plane algorithm for perfect matchings. In our algorithm, cuts separating the current optimum are easy to derive while a small LP is solved to identify the cuts that are to be retained for later iterations. Our result gives a rigorous theoretical explanation for the practical efficiency of the cutting plane approach for perfect matchings evident from implementations [40, 76]. In summary, this thesis contributes to new models and connections, new algorithms and rigorous analysis of well-known approaches for IP. ix CHAPTER I INTRODUCTION Integer Programming (IP) is a widely-used formulation for discrete optimization/combinatorial problems. In a discrete optimization problem, the variables belong to a discrete set, typically a subset of integers. This integer restriction leads to a dichotomy regarding solvability of problems: some combinatorial problems can be solved easily by exploiting the restriction, while others become very difficult to solve. Researchers in theoretical computer science and optimization have long sought to understand this dichotomy in the solvability of combinatorial problems. Most combinatorial problems involve either verifying feasibility or optimizing an objective function subject to a given set of constraints. In the feasibility version of IP, the goal n is to verify if there exists an integer point x 2 Z satisfying a given set of linear constraints m×n m Ax ≤ b, where A 2 R is a matrix and b 2 R is a vector. The linear constraints n describe the facets of a polytope P = fx 2 R : Ax ≤ bg, and thus, one would like to verify whether a given polytope contains an integer point. In the optimization version, the goal n T is to identify an integer point x 2 Z that maximizes a given objective function c x, where n c.

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