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.