New Support Size Bounds and Proximity Bounds for Integer Linear Programming

Sebastian Berndt, Matthias Mnich, Tobias Stamm

Abstract: Integer linear programming (ILP) is a fundamental research paradigm in algorithms. Many modern algorithms to solve structured ILPs efficiently follow one of two main approaches. The first one is to prove a small upper bound on the support size of the ILP, which is the number of variables taking non-zero values in an optimal solution, and then to only search for ILP solutions of small support. The second one is to apply an augmentation algorithm using Graver elements to an initial feasible solution obtained from a small proximity bound for the ILP, which is the distance between an optimal solution of the ILP and that of its LP relaxation.