Approximation algorithms for covering/packing integer programs

Given matrices A and B and vectors a, b, c and d, all with non-negative entries, we consider the problem of computing min{cx : x ∈ Z+, Ax ≥ a, Bx ≤ b, x ≤ d}. We give a bicriteria-approximation algorithm that, given ε ∈ (0, 1], finds a solution of cost O(ln(m)/ε) times optimal, meeting the covering constraints (Ax ≥ a) and multiplicity constraints (x ≤ d), and satisfying Bx ≤ (1 + ε)b + β, where β is the vector of row sums βi = ∑ j Bij. Here m denotes the number of rows of A. This gives an O(lnm)-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(maxj ∑ i Aij)) since 1982. CIP contains the set cover problem as a special case, so O(lnm)-approximation is the best possible unless P=NP.