2-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some 2-level polytopes arising in combinatorial settings. Our first contribution is proving that f0(P )f d−1 (P ) ≤ d2 d+1 for a large collection of families of such polytopes P . Here f0(P ) (resp. f d−1 (P )) is the number of vertices (resp. facets) of P , and d is its dimension. Whether this holds for all 2-level polytopes was asked in [7], and experimental results from [16] showed it true for d ≤ 7. The key to most of our proofs is a deeper understanding of the relations among those polytopes and their underlying combinatorial structures. This leads to a number of results that we believe to be of independent interest: a trade-off formula for the number of cliques and stable sets in a graph; a description of stable matching polytopes as affine projections of certain order polytopes; and a linear-size description of the base polytope of matroids that are 2-level in terms of cuts of an associated tree. IntroductionLet P ⊆ R d be a polytope. We say that P is 2-level if, for each facet F of P , all the vertices of P that are not vertices of F lie in the same translate of the affine hull of F . Equivalently, P is 2-level if and only if it has theta-rank 1 [20], or all its pulling triangulations are unimodular [50], or it has a slack matrix with entries in {0, 1} [7]. Those last three definitions appeared in papers from the semidefinite programming, statistics, and polyhedral combinatorics communities respectively, showing that 2-level polytopes naturally arise in many areas of mathematics. 2-level polytopes generalize Birkhoff [53], Hanner [28], and Hansen polytopes [29], order polytopes and chain polytopes of posets [49], spanning tree polytopes of series-parallel graphs [24], stable matching polytopes [27], some min up/down polytopes [37], and stable set polytopes of perfect graphs [10].

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Abstract.The extension complexity 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 STAB(G) be the convex hull of the stable sets of G. It is easy to see that n xc(STAB(G)) n + m. We improve both of these bounds. For the upper bound, we show that xc(STAB(G)) is O( n 2 log n ), which is an improvement when G has quadratically many edges. For the lower bound, we prove that xc(STAB(G)) is Ω(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)|.

We prove that the extension complexity of the independence polytope of every regular matroid on n elements is O(n 6 ). Past results of Wong [18] and Martin [9] on extended formulations of the spanning tree polytope of a graph imply a O(n 2 ) bound for the special case of (co)graphic matroids. However, the case of a general regular matroid was open, despite recent attempts [7,16,6].

Vulnerability metrics play a key role in the understanding of cascading failures and target/random attacks to a network. The graph fragmentation problem (GFP) is the result of a worst-case analysis of a random attack. We can choose a fixed number of individuals for protection, and a nonprotected target node immediately destroys all reachable nodes. The goal is to minimize the expected number of destroyed nodes in the network. In this paper, we address the GFP by several approaches: metaheuristics, approximation algorithms, polytime methods for specific instances, and exact methods for small instances. The computational complexity of the GFP is included in our analysis, where we formally prove that the corresponding decision version of the problem is N P-complete. Furthermore, a strong inapproximability result holds: there is no polynomial approximation algorithm with factor lower than 5/3, unless P = N P. This promotes the study of specific instances of the problem for tractability and/or exact methods in exponential time. As a synthesis, we propose new vulnerability/connectivity metrics and an interplay with game theory using a closely related combinatorial problem called component order connectivity.42 Aprile et al. / Intl. Trans. in Op. Res. 26 (2019) 41-53 system. More recently, the focus moved toward disaster management, centrality, and vulnerability metrics under random/targeted attacks (Thai and Pardalos, 2011;Mauthe et al., 2016;Gouveia and Leitner, 2017).Simulation tools were developed in order to capture a large framework of cascading failures in epidemic modeling (Marzo et al., 2017). However, under cascading failures, the system is more robust when the individuals are poorly communicated, in a strong contrast with modern connectivity theory. To the best of our knowledge, there is no simulation tool available for both apparently antipodal scenarios.The graph fragmentation problem (GFP) is the product of a worst-case analysis of a random attack under cascading failures, therefore it is suitable for pandemic analysis. However, in its minmax version, we recover a previous problem called component order connectivity (COC). The corresponding max-min version for COC is a suitable connectivity metric.The goal of this paper is to present a comprehensive analysis for the GFP, its relation with COC, and new feasible vulnerability/connectivity metrics as a synthesis. Both GFP and COC are formally presented in Section 2. Section 3 contains a comprehensive analysis for the GFP. This section covers several approaches for the problem in different subsections, such as Complexity (Subsection 3.1), Approximation algorithms (Subsection 3.2), Polytime methods for special graphs (Subsection 3.3), Exact analysis (Subsection 3.4), and Metaheuristics (Subsection 3.5). Each subsection is enriched with references for further reading. In Section 4, we discuss vulnerability/connectivity metrics suggested by GFP and COC, and a potential interplay with game theory. Finally, Section 5 summarizes the conclusions and trends for fut...