On the hardness of approximating minimization problems

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