We introduce a natural abstraction of propositional proof systems that are based on cut- ting planes. This leads to a new class of proof systems that includes many well-known meth- ods, such as Gomory-Chvátal cuts, lift-and-project cuts, Sherali-Adams cuts, or split cuts. The rank of a proof system corresponds to the number of rounds that … Read more
One main objective of research in the area of mixed-integer programming is developing cutting plane techniques to improve the solvability of mixed-integer programs (MIPs). Various cutting plane separators are typically available in MIP solvers. The large number of cutting planes generated by these separators, however, can pose a computational problem. Therefore, a sophisticated cut management … Read more
We consider bounded integer knapsacks where the weights and variable upper bounds together form a superincreasing sequence. The elements of this superincreasing knapsack are exactly those vectors that are lexicographically smaller than the greedy solution to optimizing over this knapsack. We describe the convex hull of this n-dimensional set with O(n) facets. We also establish … Read more
Split cuts are among the most important and well-understood cuts for general mixed-integer programs. In this paper we consider some recent generalizations of split cuts and compare their relative strength. More precisely, we compare the elementary closures of {split}, {cross}, {crooked cross} and general {multi-branch split cuts} as well as cuts obtained from multi-row and … Read more
We give an algorithm for testing the extremality of a large class of minimal valid functions for the two-dimensional infinite group problem. Article Download View Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. II. The Unimodular Two-Dimensional Case
In this paper, we study self-avoiding walks of a given length on a graph. We consider a formulation of this problem as a binary linear program. We analyze the polyhedral structure of the underlying polytope and describe valid inequalities. Proofs for their facial properties for certain special cases are given. In a variation of this … Read more
This paper presents a new warmstarting technique in the context of a primal-dual column generation method applied to solve a particular class of combinatorial optimization problems. The technique relies on calculating an initial point and on solving auxiliary linear optimization problems to determine the step direction needed to fully restore primal and dual feasibilities after … Read more
We give an algorithm for testing the extremality of minimal valid functions for Gomory and Johnson’s infinite group problem, that are piecewise linear (possibly discontinuous) with rational breakpoints. This is the first set of necessary and sufficient conditions that can be tested algorithmically, for deciding extremality in this important class of minimal valid functions. Article … Read more
In this paper, we study the hop-constrained survivable network design problem with reliable edges. Given a graph with non-negative edge weights and node pairs Q, the hop-constrained survivable network design problem consists of constructing a minimum weight set of edges so that the induced subgraph contains at least K edge-disjoint paths containing at most L … Read more
In recent years, branch-and-cut algorithms have become firmly established as the most effective method for solving generic mixed integer linear programs (MILPs). Methods for automatically generating inequalities valid for the convex hull of solutions to such MILPs are a critical element of branch-and-cut. This paper examines the nature of the so-called separation problem, which is … Read more