Restricted Parameter Range Promise Set Cover Problems Are Easy

Let (U,S, d) be an instance of Set Cover Problem, where U = {u1, ..., un} is a n element ground set, S = {S1, ..., Sm} is a set ofm subsets of U satisfying ⋃m i=1 Si = U and d is a positive integer. In STOC 1993M. Bellare, S. Goldwasser, C. Lund and A. Russell proved the NPhardness to distinguish the following two cases of GapSetCoverη for any constant η > 1. The Yes case is the instance for which there is an exact cover of size d and the No case is the instance for which any cover of U from S has size at least ηd. This was improved by R. Raz and S. Safra in STOC 1997 about the NP-hardness for GapSetCoverclogm for some constant c. In this paper we prove that restricted parameter range subproblem is easy. For any given function of n satisfying η(n) ≥ 1, we give a polynomial time algorithm not depending on η(n) to distinguish between YES: The instance (U,S, d) where d > 2|S| 3η(n)−1 , for which there exists an exact cover of size at most d; NO: The instance (U,S, d) where d > 2|S| 3η(n)−1 , for which any cover from S has size larger than η(n)d. Thus the large part subproblem of the NP-hard promise set cover problem is actually easy. The polynomial reduction of this restricted parameter range set cover problem is constructed by using the lattice.