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## Approximating Chromatic Sum Coloring of Bipartite Graphs in Expected Polynomial Time

#### Abstract

It is known that if complexity class P is not equal to NP the sum coloring problem cannot be approximated within 1+epsilon for some positive constant epsilon.

We consider finite, undirected graphs without loops and multiple edges.

Let G=(V,E) be a graph. By a coloring of G we mean a mapping c of V to the numbers 1,2, ..., |V| . A coloring c is proper if c(v) is not equal to c(u) whenever the vertices u and v are adjacent.

Let S(G,c) is the sum_of c(v) over all vertices v. By a chromatic sum of G we mean the number S(G)=min S(G,c) where minimum is taken over all proper colorings c of G.

The problem of finding S(G) is called the sum coloring problem.

It was shown that the sum coloring problem is NP-complete.

A graph G is called bipartite if the set of vertices of G can be partitioned into

two non-empty sets V1 and V2 such that every edge of G has one end in each of the sets.

For a number b, we say that an algorithm A approximates the chromatic sum within factor b over graphs on n vertices, if for every such graph G the algorithm A outputs a proper coloring c, such that S(G,c) is not greater than b S(G).

It is known that there exists 27/26-approximation polynomial algorithm for the chromatic SUM COLORING PROBLEM on any bipartite graph. On the other side, it was shown that here exists epsilon>0, such that there is no (1+epsilon)-approximation polynomial algorithm for the sum coloring problem on bipartite graphs, unless P is not equal to NP.

In this paper we consider the problem of developing an (1+epsilon)-approximation algorithm for the sum coloring of bipartite graphs which is polynomial in the average case for arbitrary small epsilon. We prove the existence of such algorithm.

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#### Edition

Proceedings of the Institute for System Programming, vol. 27, issue 5, 2015, pp. 191-198.

ISSN 2220-6426 (Online), ISSN 2079-8156 (Print).

DOI: 10.15514/ISPRAS-2015-27(5)-11