A Sharp Bound on the Ratio Between Optimal Integer and Fractional Covers

Abstract

The ratio between the values of optimal integer and fractional solutions to a set covering problem was shown by Johnson (Johnson, D. S. Approximation algorithms for combinatorial problems. J. Comput. System. Sci. 9 256–278.) and Lovász (Lovász, L. On the ratio of optimal integral and fractional covers. Discrete Math. 13 383–390.) to be bounded by B(d) = 1 + ln d, where d is the largest column sum. We show that if n is the number of variables, [Formula: see text] is a best possible bound on this ratio. Furthermore, for every n > 20 there is a class of problems with O(2n) constraints, for which B(n) < (2/5)B(d). We also exhibit a heuristic that is guaranteed to find an integer solution such that the ratio of its value to that of an optimal fractional solution is bounded by B(n).