No abstract

Lattice-free sets (convex subsets of R d without interior integer points) and their applications for cutting-plane methods in mixed-integer optimization have been studied in recent literature. Notably, the family of all integral lattice-free polyhedra which are not properly contained in another integral lattice-free polyhedron has been of particular interest. We call these polyhedraIt is known that, for fixed d, the family Z d -maximal integral lattice-free polyhedra is finite up to unimodular equivalence. In view of possible applications in cuttingplane theory, one would like to have a classification of this family. However, this turns out to be a challenging task already for small dimensions.In contrast, the subfamily of all integral lattice-free polyhedra which are not properly contained in any other lattice-free set, which we call R d -maximal latticefree polyhedra, allow a rather simple geometric characterization. Hence, the question was raised for which dimensions the notions of Z d -maximality and R d -maximality are equivalent. This was known to be the case for dimensions one and two. On the other hand, Nill and Ziegler (2011) showed that for dimension d ≥ 4, there exist polyhedra which are Z d -maximal but not R d -maximal. In this article, we consider the remaining case d = 3 and prove that for integral polyhedra the notions of R 3 -maximality and Z 3 -maximality are equivalent. As a consequence, the classification of all R 3 -maximal integral polyhedra by Averkov, Wagner and Weismantel (2011) contains all Z 3 -maximal integral polyhedra.

Let X be the set of integer points in some polyhedron. We investigate the smallest number of facets of any polyhedron whose set of integer points is X. This quantity, which we call the relaxation complexity of X, corresponds to the smallest number of linear inequalities of any integer program having X as the set of feasible solutions that does not use auxiliary variables. We show that the use of auxiliary variables is essential for constructing polynomial size integer programming formulations in many relevant cases. In particular, we provide asymptotically tight exponential lower bounds on the relaxation complexity of the integer points of several well-known combinatorial polytopes, including the traveling salesman polytope and the spanning tree polytope. In addition to the material in the extended abstract [7] we include omitted proofs, supporting figures, discussions about properties of coefficients in such formulations, and facts about the complexity of formulations in more general settings.

We establish that the extension complexity of the n × n correlation polytope is at least 1.5n by a short proof that is self-contained except for using the fact that every face of a polyhedron is the intersection of all facets it is contained in. The main innovative aspect of the proof is a simple combinatorial argument showing that the rectangle covering number of the unique-disjointness matrix is at least 1.5 n , and thus the nondeterministic communication complexity of the uniquedisjointness predicate is at least .58n. We thereby slightly improve on the previously best known lower bounds 1.24 n and .31n, respectively.

Consider the family of graphs without k node-disjoint odd cycles, where k is a constant. Determining the complexity of the stable set problem for such graphs G is a long-standing problem. We give a polynomial-time algorithm for the case that G can be further embedded in a (possibly non-orientable) surface of bounded genus. Moreover, we obtain polynomial-size extended formulations for the respective stable set polytopes.To this end, we show that 2-sided odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed surface. This extends the fact that odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed orientable surface (Kawarabayashi & Nakamoto, 2007).Eventually, our findings allow us to reduce the original problem to the problem of finding a minimum-cost non-negative integer circulation of a certain homology class, which turns out to be efficiently solvable in our case.