Approximability of sparse integer programs

Abstract

The main focus of this paper is a pair of new approximation algorithms for sparse integer programs. First, for covering integer programs { min cx: Ax≥b, 0≤x≤d} where A has at most k nonzeroes per row, we give a k-approximation algorithm. (We assume A, b, c, d are nonnegative.) For any k≥2 and ε>0, if P≠ NP this ratio cannot be improved to k-1-ε, and under the unique games conjecture this ratio cannot be improved to k-ε. One key idea is to replace individual constraints by others that have better rounding properties but the same nonnegative integral solutions; another critical ingredient is knapsack-cover inequalities. Second, for packing integer programs { max cx:Ax≤b, 0≤x≤d} where A has at most k nonzeroes per column, we give a 2 k k 2-approximation algorithm. This is the first polynomial-time approximation algorithm for this problem with approximation ratio depending only on k, for any k>1. Our approach starts from iterated LP relaxation, and then uses probabilistic and greedy methods to recover a feasible solution. © 2009 Springer Berlin Heidelberg.