Approximation algorithms for covering/packing integer programs

Abstract

Given matrices A and B and vectors a, b, c and d, all with non-negative entries, we consider the problem of computing $\min \{c^{T}x:x\in {\mathbb{Z}}_{+}^n,\mathit{Ax}⩾a,\mathit{Bx}⩽b,x⩽d\}$. 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 ($\mathit{Ax}⩾a$) and multiplicity constraints ($x⩽d$), and satisfying $\mathit{Bx}⩽(1+\epsilon )b+\beta$, where $\beta$ is the vector of row sums ${\beta }_i={\sum }_j{B}_{\mathit{ij}}$. 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 $\mathit{Bx}⩽b$. The previous best approximation ratio has been $O(\ln ({\max }_j{\sum }_i{A}_{\mathit{ij}}))$ since 1982. CIP contains the set cover problem as a special case, so $O(\ln m)$-approximation is the best possible unless $\mathrm{P}=\mathrm{NP}$.