On the hardness of approximating minimization problems

Abstract

We prove results indicating that it is hard to computer efficiently good approximate solutions to the Graph Coloring, Set Covering and other related minimization problems. Specifically, there is an e > 0 such that Graph Coloring cannot be approximated with ratio ne unless P=NP. Set Covering cannot be approximated with ratio clog77 for any c < 1/4 unless NP is contained in DTIME[npoly log n]. Similar results follow for related problems such as Clique Cover, Fractional Chromatic Number, Dominating Set and others.