Journal of Combinatorial Optimization

, Volume 20, Issue 4, pp 429–442 | Cite as

On the max-weight edge coloring problem

  • Giorgio Lucarelli
  • Ioannis Milis
  • Vangelis T. Paschos
Article

Abstract

We study the following generalization of the classical edge coloring problem: Given a weighted graph, find a partition of its edges into matchings (colors), each one of weight equal to the maximum weight of its edges, so that the total weight of the partition is minimized. We explore the frontier between polynomial and NP-hard variants of the problem, with respect to the class of the underlying graph, as well as the approximability of NP-hard variants. In particular, we present polynomial algorithms for bounded degree trees and star of chains, as well as an approximation algorithm for bipartite graphs of maximum degree at most twelve which beats the best known approximation ratios.

Keywords

Weighted edge coloring Polynomial algorithms Approximation algorithms 

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References

  1. Afrati FN, Aslanidis T, Bampis E, Milis I (2005) Scheduling in switching networks with set-up delays. J Comb Optim 9:49–57 MATHCrossRefMathSciNetGoogle Scholar
  2. Bojarshinov VA (2001) Edge and total coloring of interval graphs. Discrete Appl Math 114:23–28 MATHCrossRefMathSciNetGoogle Scholar
  3. Brucker P, Gladky A, Hoogeveen H, Kovalyov M, Potts C, Tautenham T, van de Velde S (1998) Scheduling a batching machine. J Sched 1:31–54 MATHCrossRefMathSciNetGoogle Scholar
  4. Crescenzi P, Deng X, Papadimitriou ChH (2001) On approximating a scheduling problem. J Comb Optim 5:287–297 MATHCrossRefMathSciNetGoogle Scholar
  5. de Werra D, Demange M, Escoffier B, Monnot J, Paschos VTh (2004) Weighted coloring on planar, bipartite and split graphs: Complexity and improved approximation. In: 15th International symposium on algorithms and computation (ISAAC’04). LNCS, vol 3341. Springer, Berlin, pp 896–907 Google Scholar
  6. Demange M, de Werra D, Monnot J, Paschos VTh (2002) Weighted node coloring: When stable sets are expensive. In: 28th Workshop on graph-theoretic concepts in computer science (WG’02). LNCS, vol 2573. Springer, Berlin, pp 114–125 CrossRefGoogle Scholar
  7. Escoffier B, Monnot J, Paschos VTh (2006) Weighted coloring: further complexity and approximability results. Inf Process Lett 97:98–103 MATHMathSciNetGoogle Scholar
  8. Finke G, Jost V, Queyranne M, Sebő A (2004) Batch processing with interval graph compatibilities between tasks. Technical report, Cahiers du laboratoire Leibniz; available at http://www-leibniz.imag.fr/NEWLEIBNIZ/LesCahiers/index.xhtml
  9. Gopal IS, Wong C (1985) Minimizing the number of switchings in a SS/TDMA system. IEEE Trans Commun 33:497–501 CrossRefGoogle Scholar
  10. Holyer I (1981) The NP-completeness of edge-coloring. SIAM J Comput 10:718–720 MATHCrossRefMathSciNetGoogle Scholar
  11. Kesselman A, Kogan K (2004) Non-preemptive scheduling of optical switches. In: 47th IEEE global telecommunications conference (GLOBECOM’04), vol 3, pp 1840–1844 Google Scholar
  12. Konig D (1916) Uber graphen und ihre anwendung auf determinantentheorie und mengenlehre. Math Ann 77:453–465 CrossRefMathSciNetGoogle Scholar
  13. Lawler EL, Labetoulle J (1978) On preemptive scheduling of unrelated parallel processors by linear programming. J Assoc Comput Mach 25:612–619 MATHMathSciNetGoogle Scholar
  14. Micali S, Vazirani VV (1980) An \({O(\sqrt{|V|}|E|)}\) algorithm for finding maximum matching in general graphs. In: 21st Annual IEEE symposium on foundations of computer science (FOCS’80), pp 17–27 Google Scholar
  15. Pemmaraju SV, Raman R (2005) Approximation algorithms for the max-coloring problem. In: 32nd international colloquium on automata, languages and programming (ICALP’05). LNCS, vol 3580. Springer, Berlin, pp 1064–1075 CrossRefGoogle Scholar
  16. Pemmaraju SV, Raman R, Varadarajan KR (2004) Buffer minimization using max-coloring. In: 15th ACM-SIAM symposium on discrete algorithms (SODA’04), pp 562–571 Google Scholar
  17. Rendl F (1985) On the complexity of decomposing matrices arising in satellite communication. Oper Res Lett 4:5–8 MATHCrossRefGoogle Scholar
  18. Vizing VG (1964) On an estimate of the chromatic class of a p-graph. Diskretn Anal 3:25–30 MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Giorgio Lucarelli
    • 1
  • Ioannis Milis
    • 1
  • Vangelis T. Paschos
    • 2
  1. 1.Department of InformaticsAthens University of Economics and BusinessAthensGreece
  2. 2.LAMSADE, CNRS UMR 7024 and Université Paris-DauphineParisFrance

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