Approximation Algorithms for Covering/Packing Integer Programs

TL;DR

This paper presents a bicriteria-approximation algorithm for covering/packing integer programs, achieving an O(ln m)-approximation for CIP, which includes set cover as a special case, improving upon the long-standing approximation ratio.

Contribution

The paper introduces a new bicriteria-approximation algorithm for CIP that achieves an O(ln m) approximation ratio, matching the best possible for set cover and improving previous bounds.

Findings

Achieves O(ln m)-approximation for CIP.

Provides a bicriteria solution satisfying constraints within epsilon and beta.

Improves upon the previous approximation ratio of O(ln(max_j sum_i A_ij)).

Abstract

Given matrices A and B and vectors a, b, c and d, all with non-negative entries, we consider the problem of computing min {c.x: x in Z^n_+, Ax > a, Bx < b, x < d}. We give a bicriteria-approximation algorithm that, given epsilon in (0, 1], finds a solution of cost O(ln(m)/epsilon^2) times optimal, meeting the covering constraints (Ax > a) and multiplicity constraints (x < d), and satisfying Bx < (1 + epsilon)b + beta, where beta is the vector of row sums beta_i = sum_j B_ij. Here m denotes the number of rows of A. This gives an O(ln m)-approximation algorithm for CIP -- minimum-cost covering integer programs with multiplicity constraints, i.e., the special case when there are no packing constraints Bx < b. The previous best approximation ratio has been O(ln(max_j sum_i A_ij)) since 1982. CIP contains the set cover problem as a special case, so O(ln m)-approximation is the best possible unless P=NP.