Extension complexity of stable set polytopes of bipartite graphs

The extension complexity $\mathsf{xc}(P)$ of a polytope $P$ is the minimum number of facets of a polytope that affinely projects to $P$. Let $G$ be a bipartite graph with $n$ vertices, $m$ edges, and no isolated vertices. Let $\mathsf{STAB}(G)$ be the convex hull of the stable sets of $G$. It is easy to see that $n \leqslant \mathsf{xc} (\mathsf{STAB}(G)) \leqslant n+m$. We improve both of these bounds. For the upper bound, we show that $\mathsf{xc} (\mathsf{STAB}(G))$ is $O(\frac{n^2}{\log n})$, which is an improvement when $G$ has quadratically many edges. For the lower bound, we prove that $\mathsf{xc} (\mathsf{STAB}(G))$ is $\Omega(n \log n)$ when $G$ is the incidence graph of a finite projective plane. We also provide examples of $3$-regular bipartite graphs $G$ such that the edge vs stable set matrix of $G$ has a fooling set of size $|E(G)|$.Comment: 13 pages, 2 figure

On some problems related to 2-level polytopes

In this thesis we investigate a number of problems related to 2-level polytopes, in particular from the point of view of the combinatorial structure and the extension complexity. 2-level polytopes were introduced as a generalization of stable set polytopes of perfect graphs, and despite their apparently simple structure, are at the center of many open problems ranging from information theory to semidefinite programming. The extension complexity of a polytope P is a measure of the complexity of representing P: it is the smallest size of an extended formulation of P, which in turn is a linear description of a polyhedron that projects down to P. In the first chapter, we examine several classes of 2-level polytopes arising in combinatorial settings and we prove a relation between the number of vertices and facets of such polytopes, which is conjectured to hold for all 2-level polytopes. The proofs are obtained through an improved understanding of the combinatorial structure of such polytopes, which in some cases leads to results of independent interest. In the second chapter, we study the extension complexity of a restricted class of 2-level polytopes, the stable set polytopes of bipartite graphs, for which we obtain non-trivial lower and upper bounds. In the third chapter we study slack matrices of 2-level polytopes, important combinatorial objects related to extension complexity, defining operations on them and giving algorithms for the following recognition problem: given a matrix, determine whether it is a slack matrix of some special class of 2-level polytopes. In the fourth chapter we address the problem of explicitly obtaining small size extended formulations whose existence is guaranteed by communication protocols. In particular we give an algorithm to write down extended formulations for the stable set polytope of perfect graphs, making a well known result by Yannakakis constructive, and we extend this to all deterministic protocols

On the linear extension complexity of stable set polytopes for perfect graphs

We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-join and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we investigate the behaviour of the extension complexity under these graph operations. We show bounds for the extension complexity of the stable set polytope of a perfect graph G depending linearly on the size of G and involving the depth of a decomposition tree of G in terms of basic perfect graphs

On the linear extension complexity of stable set polytopes for perfect graphs

We study the linear extension complexity of stable set polytopes of perfect graphs. We make use of known structural results permitting to decompose perfect graphs into basic perfect graphs by means of two graph operations: 2-joins and skew partitions. Exploiting the link between extension complexity and the nonnegative rank of an associated slack matrix, we investigate the behavior of the extension complexity under these graph operations. We show bounds for the extension complexity of the stable set polytope of a perfect graph G depending linearly on the size of G and involving the depth of a decomposition tree of G in terms of basic perfect graphs

Lifting Linear Extension Complexity Bounds to the Mixed-Integer Setting

Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer programs. In particular, prior to this work, it was open whether some classical problems, like the minimum odd-cut problem, can be expressed by a compact mixed-integer program with few (even constantly many) integer variables. This is in stark contrast to linear formulations, where recent breakthroughs in the field of extended formulations have shown that many polytopes associated to classical combinatorial optimization problems do not even admit approximate extended formulations of sub-exponential size. We provide a general framework for lifting inapproximability results of extended formulations to the setting of mixed-integer extended formulations, and obtain almost tight lower bounds on the number of integer variables needed to describe a variety of classical combinatorial optimization problems. Among the implications we obtain, we show that any mixed-integer extended formulation of sub-exponential size for the matching polytope, cut polytope, traveling salesman polytope or dominant of the odd-cut polytope, needs $\Omega(n/\log n)$ many integer variables, where $n$ is the number of vertices of the underlying graph. Conversely, the above-mentioned polyhedra admit polynomial-size mixed-integer formulations with only $O(n)$ or $O(n \log n)$ (for the traveling salesman polytope) many integer variables. Our results build upon a new decomposition technique that, for any convex set $C$, allows for approximating any mixed-integer description of $C$ by the intersection of $C$ with the union of a small number of affine subspaces.Comment: A conference version of this paper will be presented at SODA 201