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, in [BBH+96], it was shown that in general graphs this problem cannot be approximated within n~(1-#epsilon#), for any #epsilon# > 0, unless N P = Z P P. 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 = N P. 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 towards 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 results.
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机译：图的最小颜色和的问题是彩色图的顶点，使得所有分配颜色的总和（平均值）最小。最近，在[BBH + 96]中，显示在一般图中，对于任何＃epsilon＃> 0，除非NP = ZP P.在同一篇论文中，否则该问题在N〜（1-＃epsilon＃）中，此问题不能近似。 ，为二分图提供了9/8近似算法。在二角形图上为此问题的硬度问题留下了打开。在本文中，我们表明，除非p = n p，否则双方图表的最小颜色总和问题承认没有多项式近似方案，除非p = n p.证明是通过l降低确定最大程度为四个的图表中的最大独立集的问题这个问题。该结果清楚地表明，最小颜色和问题比传统着色问题更难地难以在二分形图中溶解。至于近似率，我们进一步迈向找到精确的阈值。我们提出了一种多项式10/9近似算法。除了先前结果中使用的最大独立集合之外，我们的算法还使用流程。
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