On the Maximum Edge Coloring Problem

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 present new approximation algorithms for several variants of the problem with respect to the class of the underlying graph. In particular, we deal with variants which either are known to be NP-hard (general and bipartite graphs) or are proven to be NP-hard in this paper (complete graphs with bi-valued edge weights) or their complexity question still remains open (trees).

[1]  A. Hilton,et al.  Regular Graphs of High Degree are 1‐Factorizable , 1985 .

[2]  Evripidis Bampis,et al.  Scheduling in Switching Networks with Set-Up Delays , 2005, J. Comb. Optim..

[3]  Ian Holyer,et al.  The NP-Completeness of Edge-Coloring , 1981, SIAM J. Comput..

[4]  Gerd Finke,et al.  Batch processing with interval graph compatibilities between tasks , 2005, Discret. Appl. Math..

[5]  Dominique de Werra,et al.  Restrictions and Preassignments in Preemptive open Shop Scheduling , 1996, Discret. Appl. Math..

[6]  Xiaotie Deng,et al.  On Approximating a Scheduling Problem , 2000 .

[7]  Vangelis Th. Paschos,et al.  Weighted Coloring: further complexity and approximability results , 2006, Inf. Process. Lett..

[8]  Vangelis Th. Paschos,et al.  Weighted Node Coloring: When Stable Sets Are Expensive , 2002, WG.

[9]  Chak-Kuen Wong,et al.  Minimizing the Number of Switchings in an SS/TDMA System , 1985, IEEE Trans. Commun..

[10]  Silvio Micali,et al.  An O(v|v| c |E|) algoithm for finding maximum matching in general graphs , 1980, 21st Annual Symposium on Foundations of Computer Science (sfcs 1980).

[11]  Robin J. Wilson,et al.  Edge-colourings of graphs , 1977 .

[12]  Kirill Kogan,et al.  Nonpreemptive Scheduling of Optical Switches , 2007, IEEE Transactions on Communications.

[13]  J. A. Hoogeveen,et al.  Scheduling a batching machine , 1998 .

[14]  V. Paschos,et al.  On a generalized graph coloring/batch scheduling problem , 2007 .

[15]  Rajiv Raman,et al.  Buffer minimization using max-coloring , 2004, SODA '04.

[16]  Rajiv Raman,et al.  Approximation Algorithms for the Max-coloring Problem , 2005, ICALP.

[17]  Marek Kubale.  Some results concerning the complexity of restricted colorings of graphs , 1992, Discret. Appl. Math..

[18]  F. Rendl.  On the complexity of decomposing matrices arising in satellite communication , 1985 .

[19]  D. König.  Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre , 1916 .

[20]  Shimon Even,et al.  An O (N2.5) algorithm for maximum matching in general graphs , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[21]  Vangelis Th. Paschos,et al.  Weighted Coloring on Planar, Bipartite and Split Graphs: Complexity and Improved Approximation , 2004, ISAAC.