Robert Hildebrand 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
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
Recently, cutting planes derived from maximal lattice-free convex sets have been studied intensively by the integer programming community. An important question in this research area has been to decide whether the closures associated with certain families of lattice-free sets are polyhedra. For a long time, the only result known was the celebrated theorem of Cook, … Read more
We prove that any minimal valid function for the k-dimensional infinite group relaxation that is piecewise linear with at most k+1 slopes and does not factor through a linear map with non-trivial kernel is extreme. This generalizes a theorem of Gomory and Johnson for k=1, and Cornu\’ejols and Molinaro for k=2. Article Download View A … Read more
We study a mixed integer linear program with $m$ integer variables and $k$ non-negative continuous variables in the form of the relaxation of the corner polyhedron that was introduced by Andersen, Louveaux, Weismantel and Wolsey [\emph{Inequalities from two rows of a simplex tableau}, Proc.\ IPCO 2007, LNCS, vol.~4513, Springer, pp.~1–15]. We describe the facets of … Read more
We present a faster exponential-time algorithm for integer optimization over quasi-convex polynomials. We study the minimization of a quasi-convex polynomial subject to s quasi-convex polynomial constraints and integrality constraints for all variables. The new algorithm is an improvement upon the best known algorithm due to Heinz (Journal of Complexity, 2005). A lower time complexity is … Read more