We show that there are 0-1 and unbounded knapsack polytopes with super-polynomial extension complexity. More specifically, for each n in N we exhibit 0-1 and unbounded knapsack polyhedra in dimension n with extension complexity \Omega(2^\sqrt{n}). Article Download View A NOTE ON THE EXTENSION COMPLEXITY OF THE KNAPSACK POLYTOPE

Lower bounds for Chvátal-Gomory style operators

Valid inequalities or cutting planes for (mixed-) integer programming problems are an essential theoretical tool for studying combinatorial properties of polyhedra. They are also of utmost importance for solving optimization problems in practice; in fact any modern solver relies on several families of cutting planes. The Chvátal-Gomory procedure, one such approach, has a peculiarity that … Read more

Design and Verify: A New Scheme for Generating Cutting-Planes

A cutting-plane procedure for integer programming (IP) problems usually involves invoking a black-box procedure (such as the Gomory-Chvatal (GC) procedure) to compute a cutting-plane. In this paper, we describe an alternative paradigm of using the same cutting-plane black-box. This involves two steps. In the first step, we design an inequality cx = d + 1\} … Read more

Integer-empty polytopes in the 0/1-cube with maximal Gomory-Chvátal rank

We provide a complete characterization of all polytopes P ⊆ [0,1]^n with empty integer hull whose Gomory-Chvátal rank is n (and, therefore, maximal). In particular, we show that the first Gomory- Chvátal closure of all these polytopes is identical. Article Download View Integer-empty polytopes in the 0/1-cube with maximal Gomory-Chvátal rank

Random half-integral polytopes

We show that half-integral polytopes obtained as the convex hull of a random set of half-integral points of the 0/1 cube have rank as high as Ω(logn/loglogn) with positive probability — even if the size of the set relative to the total number of half-integral points of the cube tends to 0. The high rank … Read more

Lower bounds for the Chvátal-Gomory rank in the 0/1 cube

Although well studied, important questions on the rank of the Chvátal-Gomory operator when restricting to polytopes contained in the n-dimensional 0/1 cube have not been answered yet. In particular, the question on the maximal rank of the Chvátal-Gomory procedure for this class of polytopes is still open. So far, the best-known upper bound is O(n^2 … Read more

Symmetry-exploiting cuts for a class of mixed-0/1 second order cone programs

We will analyze mixed 0/1 second order cone programs where the fractional and binary variables are solely coupled via the conic constraints. For this special type of mixed-integer second order cone programs we devise a cutting-plane framework based on the generalized Benders cut and an implicit Sherali-Adams reformulation. We show that the resulting cuts are … Read more

On the connection of the Sherali-Adams closure and border bases

The Sherali-Adams lift-and-project hierarchy is a fundamental construct in integer programming, which provides successively tighter linear programming relaxations of the integer hull of a polytope. We initiate a new approach to understanding the Sherali-Adams procedure by relating it to methods from computational algebraic geometry. Our main result is a refinement of the Sherali-Adams procedure that … Read more