Tight Approximation Results for General Covering Integer Programs

In this paper we study approximation algorithms for solving a general covering integer program. An n-vector x of nonnegative integers is sought, which minimizes cT x; subject to Ax \geqslant b,x \leqslant d: The entries of A; b; c are nonnegative. Let m be the number of rows of A: Covering problems have been heavily studied in combinatorial optimization. We focus on the effect of the multiplicity constraints, x \leqslant d; on approximability. Two longstanding open questions remain for this general formulation with upper bounds on the variables.(i) The integrality gap of the standard LP relaxation is arbitrarily large. Existing approximation algorithms that achieve the well-known O(logm)-approximation with respect to the LP value do so at the expense of violating the upper bounds on the variables by the same O(logm) multiplicative factor. What is the smallest possible violation of the upper bounds that still achieves cost within O(logm) of the standard LP optimum__ __(ii) The best known approximation ratio for the problem has been 0(\log (\max _j \sum\nolimits_i {A_{ij} })) since 1982. This bound can be as bad as polynomial in the input size. Is an O(logm)-approximation, like the one known for the special case of Set Cover, possible__ __We settle these two open questions. To answer the first question we give an algorithm based on the relatively simple new idea of randomly rounding variables to smaller-than integerunits. To settle the second question we give a reduction from approximating the problem while respecting multiplicity constraints to approximating the problem with a bounded violation of the multiplicity constraints.