TY - GEN

T1 - The minimum color sum of bipartite graphs

AU - Bar-Noy, Amotz

AU - Kortsarz, Guy

N1 - Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1997.
Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 1997

Y1 - 1997

N2 - 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 n1-є, for any є > 0, unless NP = ZPP. 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 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.

AB - 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 n1-є, for any є > 0, unless NP = ZPP. 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 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.

UR - http://www.scopus.com/inward/record.url?scp=84951060794&partnerID=8YFLogxK

U2 - 10.1007/3-540-63165-8_227

DO - 10.1007/3-540-63165-8_227

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AN - SCOPUS:84951060794

SN - 3540631658

SN - 9783540631651

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 738

EP - 748

BT - Automata, Languages and Programming - 24th International Colloquium, ICALP 1997, Proceedings

A2 - Degano, Pierpaolo

A2 - Gorrieri, Roberto

A2 - Marchetti-Spaccamela, Alberto

PB - Springer Verlag

Y2 - 7 July 1997 through 11 July 1997

ER -