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A Combinatorial Algorithm for Minimum Weighted Colorings of Claw-Free Perfect Graphs

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Abstract

We present an O(n5) combinatorial algorithm for the minimum weighted coloring problem on claw-free perfect graphs, which was posed by Hsu and Nemhauser in 1984. Our algorithm heavily relies on the structural descriptions of claw-free perfect graphs given by Chavátal and Sbihi and by Maffray and Reed.

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References

  • C. Berge and V. Chvátal (Eds.), Topics on Perfect Graphs. North-Holland, Amsterdam, 1984.

    Google Scholar 

  • M. Chudnovsky, N. Robertson, P.D. Seymour, and R. Thomas, “The strong perfect graph theorem,” Ann. of Math., to appear.

  • V. Chvátal, Linear Programming. W.H. Freeman and Company: New York, 1983.

    Google Scholar 

  • V. Chvátal and N. Sbihi, Bull-free Berge graphs are perfect. Graphs Combin, vol. 3, pp. 127–139, 1987.

    Article  Google Scholar 

  • V. Chvátal and N. Sbihi, “Recognizing claw-free perfect graphs,” J. Combin. Theory Ser. B, vol. 44, pp. 154–176, 1988.

    Article  Google Scholar 

  • D.G. Degiorgi and K. Simon, “A dynamic algorithm for line graph recognition,” Lecture Notes in Comput. Sci., vol. 1017, pp. 37–48, 1995.

    Google Scholar 

  • L.R. Ford, Jr. and D.R. Fulkerson, “Maximal flow through a network,” Canad. J. Math., vol. 8, pp. 399–404, 1956.

    Google Scholar 

  • M.R. Garey and D.S. Johnson, Computers and Intractability. W.H. Freeman and Company, New York, 1979.

    Google Scholar 

  • F. Gavril, “Algorithms on clique separable graphs,” Discrete Math., vol. 55, pp. 245–254, 1985.

    Article  Google Scholar 

  • T. Gonzalez and S. Sahni, “Open shop scheduling to minimize finish time,” J. Assoc. Comput. Mach., vol. 23, pp. 665–679, 1976.

    Google Scholar 

  • M. Grötschel, L. Lovász, and A. Schrijver, “Polynomial algorithms for perfect graphs,” in Topics on Perfect Graphs C. Berge and V. Chvátal (Eds.) North-Holland, Amsterdam, 1984, pp. 325–356

    Google Scholar 

  • J.E. Hopcroft and R.M. Karp, “An n5/2 algorithm for maximum matchings in bipartite graphs,” SIAM J. Comput., vol. 2, pp. 225–231, 1973.

    Article  Google Scholar 

  • W.L. Hsu, “How to color claw-free perfect graphs,” Ann. Discrete Math., vol. 11, pp. 189–197, 1981.

    Google Scholar 

  • W.L. Hsu and G. Nemhauser, “Algorithms for maximum weight cliques, minimum weighted clique covers and minimum colorings of claw-free perfect graphs,” in Topics on Perfect Graphs C. Berge and V. Chvátal (Eds.) North-Holland, Amsterdam, 1984, pp. 357–369.

    Google Scholar 

  • W.L. Hsu and G. Nemhauser, “Algorithms for minimum covering by cliques and maximum clique in claw-free perfect graphs,” Discrete Math., vol. 37, pp. 181–191, 1981.

    Article  Google Scholar 

  • W.L. Hsu and G. Nemhauser, “A polynomial algorithm for the minimum weighted clique cover problem on claw-free perfect graphs,” Discrete Math., vol. 38, pp. 65–71, 1982.

    Article  Google Scholar 

  • A.V. Karzanov, “Determining the maximal flow in a network by the method of preflows,” Soviet Math. Dokl., vol. 15, pp. 434–437, 1974.

    Google Scholar 

  • P.G.H. Lehot, “An optimal algorithm to detect a line graph and output its root graph,” J. Assoc. Comput. Mach., vol. 21, pp. 569–575, 1974.

    Google Scholar 

  • L. Lovász, Normal hypergraphs and the perfect graph conjecture,” Discrete Math., vol. 2, pp. 253–267, 1972.

    Article  Google Scholar 

  • F. Maffray and B. Reed, A description of claw-free perfect graphs,” J. Combin. Theory Ser. B, vol. 75, pp. 134–156, 1999.

    Article  Google Scholar 

  • K.R. Parthasarathy and G. Ravindra, “The strong perfect graph conjecture is true for K1,3-free graphs,” J. Combin. Theory Ser. B, vol. 21, pp. 212–223, 1976.

    Google Scholar 

  • J.L. Ramfrez-Alfonsín and B.A. Reed (Eds.), Perfect Graphs. John Wiley & Sons, Ltd., Chichester, 2001.

    Google Scholar 

  • N.D. Roussopoulos, “A m, n algorithm for determining the graph H from its line graph G,” Inform. Process. Lett., vol. 2, pp. 108–112, 1973.

    Article  Google Scholar 

  • A. Schrijver, Theory of Linear and Integer Programming. John Wiley & Sons, 1986.

  • R.E. Tarjan, “Decomposition by clique separators,” Discrete Math., vol. 55, pp. 221–232, 1985.

    Article  Google Scholar 

  • R.E. Tarjan, Data Structures and Network Algorithms, SIAM, Philadelphia, 1983.

    Google Scholar 

  • S.H. Whitesides, “An algorithm for finding clique cut-sets,” Inform. Process. Lett., vol. 12, pp. 31–32, 1981.

    Article  Google Scholar 

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Authors and Affiliations

  1. Center for Combinatorics, Nankai University, Tianjin, China

    Xueliang Li

  2. Department of Mathematics, University of Hong Kong, Hong Kong, China

    Wenan Zang

Authors
  1. Xueliang Li
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  2. Wenan Zang
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Correspondence to Wenan Zang.

Additional information

Supported by the National Science Foundation of China.

Supported by the Research Grants Council of Hong Kong (Project No. HKU 7109/01P) and Seed Funding for Basic Research of HKU.

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Li, X., Zang, W. A Combinatorial Algorithm for Minimum Weighted Colorings of Claw-Free Perfect Graphs. J Comb Optim 9, 331–347 (2005). https://doi.org/10.1007/s10878-005-1775-y

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  • DOI: https://doi.org/10.1007/s10878-005-1775-y

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