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
T2 - 24th International Colloquium on Automata, Languages and Programming, ICALP 1997
Y2 - 7 July 1997 through 11 July 1997
ER -