Abstract
We study the complexity of the following problem that we call Max edge t-coloring: given a multigraph G and a parameter t, color as many edges as possible using t colors, such that no two adjacent edges are colored with the same color. (Equivalently, find the largest edge induced subgraph of G that has chromatic index at most t). We show that for every fixed t ≥ 2 there is some ∈ > 0 such that it is NP-hard to approximate Max edge t-coloring within a ratio better than 1-∈. We design approximation algorithms for the problem with constant factor approximation ratios. An interesting feature of our algorithms is that they allow us to estimate the value of the optimum solution up to a multiplicative factor that tends to 1 as t grows. Our study was motivated by call admittance issues in satellite based telecommunication networks.
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References
L. D. Andersen: On edge-colourings of graphs. Math. Scand. 40:161–175, 1977.
Alberto Caprara and Romeo Rizzi: Improving a Family of Approximation Algorithms to Edge Color Multigraphs. Information Processing Letters, 68:11–15, 1998.
R. Cole and K. Ost and S. Schirra: Edge-coloring Bipartite Multigraphs in O(E log D) Time. Combinatorica, 21(1), 2001.
Gerard Cornuejols and William Pulleyblank: A matching Problem with Side Conditions. Discrete Mathematics, 29:135–159,1980.
Reinhard Diestel: Graph Theory. Springer 1996.
J. Edmonds: Maximum matching and a polyhedron with 0, 1 vertices. Journal of Research National Bureau of Standards, 69B:125–130, 1965.
J. Edmonds: Paths, trees, and flowers. Canadian Journal of Mathematics, 17:449–467, 1965.
Lene M. Favrholdt and Morten N. Nielsen: On-Line Edge-Coloring with a Fixed Number of Colors. Foundations of Software Technology and Theoretical Computer Science 20, 2000.
U. Feige and E. Ofek and U. Wieder: Approximating maximum edge coloring in multigraphs. Technical report, Weizmann Institute, 2002.
M.K. Goldberg: On multigraphs of almost maximal choratic class (in Russion). Diskret. Analiz, 1973.
M. Grotschel and L. Lovasz and A. Schrijver: Geometric algorithms and combinatorial optimization. Springer-Verlag, Berlin, 1988.
D. Hartvigsen: Extensions of Matching Theory. Carnegie-Mellon University, 1984.
Dorit Hochbaum: Approximation Algorithms For NP-Hard Problems. PWS Publishing Company, Boston, 1997.
Ian Holyer: The NP-completeness of edge-coloring. SIAM Journal on Computing, 10(4):718–720, 1981.
J. Kahn: Asymptotics of the Chromatic Index for Multigraphs. Journal of Combinatorial Theory, Series B, 68, 1996.
Daniel Leven and Zvi Galil: NP Completeness of Finding the Chromatic Index of Regular Graphs. Journal of Algorithms, 4(1):35–44, 1983.
L. Lovasz and Plummer:Matching Theory, Annals of Discrete Mathematics 29 North-Holland, 1986.
Takao Nishizeki and Kenichi Kashiwagi: On the 1.1 Edge-Coloring of Multigraphs. SIAM Journal on Discrete Mathematics 3:391–410, 1990.
T. Nishizeki and X. Zhou: Edge-Coloring and f-coloring for various classes of graphs. Journal of Graph Algorithms and Applications, 3, 1999.
Eran Ofek: Maximum edge coloring with a bounded number of colors. Weizmann Institute of Science, Rehovot, Israel, November 2001.
M. Padberg and M. R. Rao: Odd minimum cut-sets and b-matchings. Mathematics of Operations Research, 1982, 7:67–80.
Christos H. Papadimitriou and Mihalis Yannakakis: Optimization, Approximation, and Complexity Classes. Journal of Computer and System Sciences, 43(3):425–440, 1991.
P. Seymour: Some unsolved problems on one-factorizations of graphs. Graph Theory and Related Topics, Academic Press, 367–368, 1979.
V. G. Vizing: On an estimate of the chromatic class of a p-graph (in Russian). Diskret. Analiz 3:23–30, 1964.
Ehud Wieder: Offine satellite resource allocation or Maximum edge coloring with a fixed number of colors.Weizmann Institute of Science, Rehovot, Israel, November 2001.
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Feige, U., Ofek, E., Wieder, U. (2002). Approximating Maximum Edge Coloring in Multigraphs. In: Jansen, K., Leonardi, S., Vazirani, V. (eds) Approximation Algorithms for Combinatorial Optimization. APPROX 2002. Lecture Notes in Computer Science, vol 2462. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45753-4_11
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DOI: https://doi.org/10.1007/3-540-45753-4_11
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