We introduce a new class of valid inequalities for any 0/1 knapsack polytope, called Solitary item inequality, which are facet-defining. We prove that any facet-defining inequality of a 0/1 knapsack polytope with nonnegative integral coefficients and right hand side 1 belongs to this class, and hence, the set of facet-defining inequalities corresponding to strong covers … Read more
We propose two new classes of valid inequalities (VIs) for the binary knapsack polytope, based on non-minimal covers. We also show that these VIs can be obtained through neither sequential nor simultaneous lifting of well-known cover inequalities. We further provide conditions under which they are facet-defining. The usefulness of these VIs is demonstrated using computational … Read more
DP is a complexity class that is the class of all languages that are the intersection of a language in NP and a language in co-NP, as coined by Papadimitriou and Yannakakis. In this paper, we will establish that, recognizing a facet for the knapsack polytope is DP-complete, as conjectured by Hartvigsen and Zemel in … Read more
We close three open problems in the separation complexity of valid inequalities for the knapsack polytope. Specifically, we establish that the separation problems for extended cover inequalities, (1,k)-configuration inequalities, and weight inequalities are all NP-complete. We also give a number of special cases where the separation problem can be solved in polynomial time. ArticleDownload View … Read more
We propose a method to generate cutting-planes from multiple covers of knapsack constraints. The covers may come from different knapsack inequalities if the weights in the inequalities form a totally-ordered set. Thus, we introduce and study the structure of a totally-ordered multiple knapsack set. The valid multi-cover inequalities we derive for its convex hull have … Read more
A desirable property of integer formulations is to consist of few inequalities having small coefficients. We show that these targets are conflicting by proving the existence of knapsack sets that need exponentially many inequalities or exponentially large coefficients in any integer formulation. Moreover, we show that there exist undirected graphs such that (in a natural … Read more
The 0/1 knapsack polytope is the convex hull of all 0/1 vectors that satisfy a given single linear inequality with non-negative coefficients. This paper provides a comprehensive overview of knapsack polytopes. We discuss basic polyhedral properties, (lifted) cover and other valid inequalities, cases for which complete linear descriptions are known, geometric properties for small dimensions, … Read more
This paper investigates a polyhedral approach to handle symmetries in mixed-binary programs. We study symretopes, i.e., the convex hulls of all binary vectors that are lexicographically maximal in their orbit with respect to the symmetry group. These polytopes turn out to be quite complex. For practical use, we therefore develop an integer programming formulation with … Read more
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}). ArticleDownload View PDF