Approximability of Sparse Integer Programs

TL;DR

This paper introduces new approximation algorithms for sparse integer programs, providing near-optimal ratios under complexity assumptions, and establishes inapproximability bounds for certain cases.

Contribution

The paper presents novel approximation algorithms for covering and packing integer programs with sparsity constraints, improving known bounds and introducing new techniques like constraint replacement and knapsack-cover inequalities.

Findings

A k-approximation for covering integer programs with at most k nonzeroes per row.

A (2k^2+2)-approximation for packing integer programs with at most k nonzeroes per column.

An inapproximability bound of 17/16 for covering integer programs with at most two nonzeroes per column.

Abstract

The main focus of this paper is a pair of new approximation algorithms for certain 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 eps>0, if P != NP this ratio cannot be improved to k-1-eps, and under the unique games conjecture this ratio cannot be improved to k-eps. 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 (2k^2+2)-approximation algorithm. Our approach builds on the iterated LP relaxation framework. In addition, we obtain improved approximations for the second problem when k=2, and for both problems when every A_{ij} is small compared to b_i. Finally, we demonstrate a 17/16-inapproximability for covering integer programs with at most two nonzeroes per column.