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Minimum Color Sum of Bipartite Graphs

✍ Scribed by Amotz Bar-Noy; Guy Kortsarz

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 242 KB
Volume: 28
Category: Article
ISSN: 0196-6774
DOI: 10.1006/jagm.1998.0938

No coin nor oath required. For personal study only.

✦ Synopsis

The problem of minimum color sum of a graph is to color the vertices of the Ž . 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 1y ⑀ Ž n , for any ⑀ ) 0, unless NP s ZPP Bar-Noy et al., Information and Computa-Ž .

. tion 140 1998 , 183᎐202 . In the same paper, a 9r8-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 s 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 10r9-approximation algorithm. Our algorithm uses a flow procedure in addition to the maximum independent set procedure used in previous solutions.

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