Title
Optimal edge-coloring with edge rate constraints.
Abstract
We consider the problem of covering the edges of a graph by a sequence of matchings subject to the constraint that each edge e appears in at least a given fraction r(e) of the matchings. Although it can be determined in polynomial time whether such a sequence of matchings exists or not [Grotschel et al., Combinatorica (1981), 169-197], we show that several questions about the length of the sequence are computationally intractable. Therefore, as is commonly done [Golumbic, Algorithmic graph theory and perfect graphs, 2004], we restrict our investigation to a special class of graphs. In recent work [Birand et al., INFOCOM 2010 Proceedings, 2010], two of the authors dealt with so-called OLoP (Overall Local Pooling) graphs, a class of graphs for which similar matching-related problems are tractable (namely, in an online distributed wireless network scheduling setting). We therefore focus on these graphs and generalize the results to a larger class of graphs which we call GOLoP graphs. In particular, we show that deciding whether a given GOLoP graph has a matching sequence of length at most k can be done in linear time. In case the answer is affirmative, we show how to construct, in quadratic time, the matching sequence of length at most k. Finally, we prove that, for GOLoP graphs, the length of a shortest sequence does not exceed a constant times the least common denominator of the fractions r(e), leading to a pseudopolynomial-time algorithm for minimizing the length of the sequence. We show that the constant equals 1 for OLoP graphs and, following Seymour [Seymour, Proc. London Math. Soc., 1979], conjecture that the constant is as small as 2 for general graphs. We then show that this conjecture holds for all graphs with at most 10 vertices. (c) 2013 Wiley Periodicals, Inc. NETWORKS, Vol 62(3), 165-182 2013
Year
DOI
Venue
2013
10.1002/net.21505
NETWORKS
Keywords
Field
DocType
edge coloring,fractional edge coloring,nearly bipartite graphs,scheduling algorithms,throughput maximization,greedy maximal scheduling,wireless networks
Discrete mathematics,Combinatorics,Mathematical optimization,Indifference graph,Chordal graph,Cograph,Pathwidth,1-planar graph,Mathematics,Trapezoid graph,Graph coloring,Strong perfect graph theorem
Journal
Volume
Issue
ISSN
62.0
3.0
0028-3045
Citations 
PageRank 
References 
0
0.34
11
Authors
4
Name
Order
Citations
PageRank
Dariusz Dereniowski117826.76
Wieslaw Kubiak254062.61
Bernard Ries317628.68
Yori Zwols4456.34