## Abstract

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-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.

Original language | English |
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Title of host publication | Automata, Languages and Programming - 24th International Colloquium, ICALP 1997, Proceedings |

Editors | Pierpaolo Degano, Roberto Gorrieri, Alberto Marchetti-Spaccamela |

Publisher | Springer Verlag |

Pages | 738-748 |

Number of pages | 11 |

ISBN (Print) | 3540631658, 9783540631651 |

DOIs | |

State | Published - 1997 |

Event | 24th International Colloquium on Automata, Languages and Programming, ICALP 1997 - Bologna, Italy Duration: 7 Jul 1997 → 11 Jul 1997 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1256 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 24th International Colloquium on Automata, Languages and Programming, ICALP 1997 |
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Country/Territory | Italy |

City | Bologna |

Period | 7/07/97 → 11/07/97 |

### Bibliographical note

Publisher Copyright:© Springer-Verlag Berlin Heidelberg 1997.