Minimum Color Sum of Bipartite Graphs

The problem of minimum color sum of a graph is to color the vertices of the graph such that the sum (average) of all assigned colors is minimum. Recently it was shown that in general graphs this problem cannot be approximated within n^1-ε, for any ε > 0, unless NP = ZPP (Bar-Noy et al., Information and Computation 140 (1998), 183-202). In the same paper, a 9/8-approximation algorithm was presented for bipartite graphs. The hardness question for this problem on bipartite graphs was left open. In this paper we show that the minimum color sum problem for bipartite graphs admits no polynomial approximation scheme, unless P = NP. The proof is by L-reducing the problem of finding the maximum independent set in a graph whose maximum degree is four to this problem. This result indicates clearly that the minimum color sum problem is much harder than the traditional coloring problem, which is trivially solvable in bipartite graphs. As for the approximation ratio, we make a further step toward finding the precise threshold. We present a polynomial 10/9-approximation algorithm. Our algorithm uses a flow procedure in addition to the maximum independent set procedure used in previous Solutions.