Cutting Planes and Integrality of Polyhedra: Structure and Complexity by Dabeen Lee A thesis submitted to Carnegie Mellon University in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Algorithms, Combinatorics, and Optimization Pittsburgh, Pennsylvania, USA, 2019 c Dabeen Lee 2019 Examining Committee Membership The following served on the Examining Committee for this thesis. External Examiners: William J. Cook, Professor Department of Applied Mathematics and Statistics Johns Hopkins University Sanjeeb Dash, Research Staff Member Department of Mathematical Sciences IBM Thomas J. Watson Research Center Supervisor: G´erardCornu´ejols,Professor Tepper School of Business Carnegie Mellon University Internal Members: Anupam Gupta, Professor School of Computer Science Carnegie Mellon University R Ravi, Professor Tepper School of Business Carnegie Mellon University ii Author's Declaration This thesis consists of material all of which I authored or co-authored: see Statement of Contributions included in the thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Statement of Contributions This thesis is based on various collaborations with Ahmad Abdi, G´erardCornu´ejols,Sanjeeb Dash, Oktay G¨unl¨uk,Nat´aliaGuri˘canov´aand Yanjun Li. These include, but are not limited to, papers [2,4,5,6, 36, 37, 53, 91]. iv Abstract In this thesis, we study theoretical aspects of integer linear programming. This thesis consists of two main parts: the first part is on the theory of cutting planes for integer linear programming, while the second part is on the theory of ideal clutters in combinatorial optimization. Cutting planes for an integer linear program are linear inequalities that are valid for all integer feasible solutions but possibly violated by some solutions to the linear programming relaxation. The Chv´atal- Gomory cuts, introduced by Gomory in 1958 and further studied by Chv´atalin 1973 in relation to their applications in combinatorial optimization, are one of the simplest types of cutting planes. The split cuts are another class of cutting planes that are important in modern integer linear programming. The first part of this thesis discusses our recent developments in the theory of Chv´atal-Gomorycuts and split cuts. We study rational polyhedra with Chv´atalrank 1, rational polyhedra with split rank 1, some sufficient conditions under which a rational polytope in the 0,1 hypercube has small Chv´atalrank, and a generalization of the Chv´atalclosure. Let E be a finite set of elements, and let C be a family of subsets of E called members. We say that C is a clutter over ground set E if no member contains another. We say that the clutter C is ideal if the P system (xe : e 2 C) ≥ 1 8C 2 C; xe ≥ 0 8e 2 E defines an integral polyhedron. One can find rich classes of ideal clutters that arise in combinatorial optimization: the clutter of st-paths, the clutter of T -joins, the clutter of dijoins, the clutter of the odd circuits of a weakly bipartite graph, etc. As these wide range of examples suggest, characterizing when a clutter is ideal is still a major open question in integer programming and combinatorial optimization. One of the conjectures that were made to understand the question is the τ = 2 Conjecture by Cornu´ejols,Guenin, and Margot in 2000. In the second part of this thesis, we study and develop tools to solve the τ = 2 Conjecture. We introduce intersecting clutters and multipartite clutters and study two equivalent versions of the τ = 2 Conjecture stated in terms of intersecting clutters and multipartite clutters. v Acknowledgements First of all, I would like to thank my Ph.D. advisor G´erardCornu´ejolsfor his continuous support, patience, and encouragement. Without his support, I would not have been able to finish this long Ph.D. journey. I am always grateful and feel fortunate to have G´erardas my advisor, and I have been happy to learn integer programming and combinatorial optimization from him and work on exciting projects together. I must thank Ahmad Abdi, who has always been an inspiring colleague and a close friend. Ahmad dragged me into the field of ideal clutters, and he has patiently taught me the theory of ideal clutters from basics to modern research questions. I am lucky to collaborate with Ahmad, and joint works with him led to one half of this thesis. I am also greatly indebted to Sanjeeb Dash and Oktay G¨unl¨ukat IBM T.J. Watson Research Center for the great internship opportunity, and I am lucky to have them as my mentors. I would also like to thank coauthors Nat´aliaGuri˘canov´aand Yanjun Li. It was great 5 years in Pittsburgh with my classmates at CMU, Michael Anastos, Gerdus Benade, Nam Ho-Nguyen, and Ryo Kimura. Lastly, I want to thank the members of Korean Central Church of Pittsburgh for including me as part of the church and for being a family to me. vi Dedication I dedicate this thesis to my parents. vii Table of Contents 1 Introduction 1 1.1 Preliminaries: the Chv´atalclosure and the split clousre....................3 1.2 Rational polytopes with rank 1..................................4 1.3 Polytopes in the 0,1 hypercube that have small Chv´atalrank.................6 1.4 Generalized Chv´atalclosure....................................8 1.5 Preliminaries: ideal clutters....................................9 1.6 Intersecting restrictions in clutters................................ 13 1.7 Multipartite clutters and the τ = 2 Conjecture......................... 16 1.8 Multipartite clutters of bounded degree............................. 19 1.9 The reflective product....................................... 20 1.10 Ideal vector spaces......................................... 22 2 Polytopes with Chvatal rank 1 26 2.1 Easy cases.............................................. 27 2.2 Recognizing rational polytopes with an empty Chv´atalclosure is NP-hard.......... 32 2.2.1 The case of polytopes contained in the unit hypercube................. 32 2.2.2 The case of simplices.................................... 36 2.2.3 Optimization and separation over Chv´atalclosure................... 42 2.2.4 Deciding whether adding a certain number of Chv´atal-Gomorycuts can yield the integer hull......................................... 42 2.3 Flatness theorem for closed convex sets with empty Chv´atalclosure............. 44 2.3.1 Flatness result....................................... 46 2.3.2 Proof of Theorem 2.19................................... 48 2.3.3 A Lenstra-type algorithm................................. 51 viii 3 Polytopes with split rank 1 53 3.1 Deciding whether the split closure of a rational polytope is empty is NP-hard........ 53 3.1.1 Reduction from Equality Knapsack............................ 54 3.1.2 Implications......................................... 58 3.2 Flatness theorem for rational polytopes of split rank 1..................... 59 3.3 Further notes............................................ 61 4 Polytopes in the 0,1 hypercube that have small Chv´atalrank 62 4.1 Basic tools.............................................. 63 4.2 The Chv´atalrank of QS ...................................... 64 4.2.1 Chv´atalrank 1....................................... 64 4.2.2 Chv´atalrank 2....................................... 65 4.2.3 Chv´atalrank 3....................................... 66 4.2.4 Chv´atalrank 4....................................... 68 4.3 Vertex cutsets............................................ 73 4.3.1 Cut vertex......................................... 73 4.3.2 2-vertex cut......................................... 76 4.4 Graphs of tree width 2....................................... 80 4.5 Proof of Theorem 1.14....................................... 84 5 Generalized Chv´atalclosure 86 5.1 Preliminaries............................................ 87 5.2 S-Chv´atalclosure for finite number of integer points...................... 90 5.3 Integer points in a cylinder.................................... 91 5.4 Integer points with bounds on components............................ 96 5.4.1 Covering polyhedra..................................... 99 5.4.2 Packing polyhedra..................................... 109 5.5 Proof of Theorem 1.16....................................... 114 5.6 Further notes............................................ 116 6 Intersecting restrictions in clutters 117 6.1 Finding an intersecting restriction................................ 117 6.2 Finding a delta and the blocker of an extended odd hole minor................ 120 6.3 Further notes............................................ 125 ix 7 Multipartite clutters 126 7.1 Multipartite clutters and the τ = 2 Conjecture......................... 127 7.2 Induced clutters.......................................... 130 7.3 Multipartite clutters of bounded degree............................. 133 7.4 A pseudocode to generate strictly polar multipartite clutters that do not pack........ 139 7.5 Further notes............................................ 143 8 The reflective product 144 8.1 Products and coproducts of clutters............................... 145 8.2 Products and reflective products of sets............................. 146 8.3 Minimally non-packing multipartite clutters obtained by the reflective product....... 148 8.4 Further notes............................................ 155 9 Ideal vector spaces 157 9.1 Theorem 1.50 for when the characteristic of GF (q) is not 2.................