The matching polytope does not admit fully-polynomial size relaxation schemes

The groundbreaking work of Rothvo{\ss} [arxiv:1311.2369] established that every linear program expressing the matching polytope has an exponential number of inequalities (formally, the matching polytope has exponential extension complexity). We generalize this result by deriving strong bounds on the polyhedral inapproximability of the matching polytope: for fixed $0 < \varepsilon < 1$, every polyhedral $(1 + \varepsilon / n)$-approximation requires an exponential number of inequalities, where $n$ is the number of vertices. This is sharp given the well-known $\rho$-approximation of size $O(\binom{n}{\rho/(\rho-1)})$ provided by the odd-sets of size up to $\rho/(\rho-1)$. Thus matching is the first problem in $P$, whose natural linear encoding does not admit a fully polynomial-size relaxation scheme (the polyhedral equivalent of an FPTAS), which provides a sharp separation from the polynomial-size relaxation scheme obtained e.g., via constant-sized odd-sets mentioned above. Our approach reuses ideas from Rothvo{\ss} [arxiv:1311.2369], however the main lower bounding technique is different. While the original proof is based on the hyperplane separation bound (also called the rectangle corruption bound), we employ the information-theoretic notion of common information as introduced in Braun and Pokutta [http://eccc.hpi-web.de/report/2013/056/], which allows to analyze perturbations of slack matrices. It turns out that the high extension complexity for the matching polytope stem from the same source of hardness as for the correlation polytope: a direct sum structure.Comment: 21 pages, 3 figure

The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution

International audienceThe Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The first of these three approaches yielded the first (and to date only) proof of the SPGC; the other two remain promising to consider in attempting an alternative proof. This paper is an unbalanced survey of the attempts to solve the SPGC; unbalanced, because (1) we devote a signicant part of it to the 'primitive graphs and structural faults' paradigm which led to the Strong Perfect Graph Theorem (SPGT); (2) we briefly present the other "direct" attempts, that is, the ones for which results exist showing one (possible) way to the proof; (3) we ignore entirely the "indirect" approaches whose aim was to get more information about the properties and structure of perfect graphs, without a direct impact on the SPGC. Our aim in this paper is to trace the path that led to the proof of the SPGT as completely as possible. Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin and B.A. Reed, eds., Perfect Graphs (Wiley & Sons, 2001).], but it also implies a deeper analysis (with additional results) and another viewpoint on the topic

A polyhedral approach for the Equitable Coloring Problem

In this work we study the polytope associated with a 0,1-integer programming formulation for the Equitable Coloring Problem. We find several families of valid inequalities and derive sufficient conditions in order to be facet-defining inequalities. We also present computational evidence that shows the efficacy of these inequalities used in a cutting-plane algorithm

Zero-one IP problems: Polyhedral descriptions & cutting plane procedures

A systematic way for tightening an IP formulation is by employing classes of linear inequalities that define facets of the convex hull of the feasible integer points of the respective problems. Describing as well as identifying these inequalities will help in the efficiency of the LP-based cutting plane methods. In this report, we review classes of inequalities that partially described zero-one poly topes such as the 0-1 knapsack polytope, the set packing polytope and the travelling salesman polytope. Facets or valid inequalities derived from the 0-1 knapsack and the set packing polytopes are algorithmically identifie