Improved Approximation Algorithms for the Max-Edge Coloring Problem

The Max Edge-Coloring (MEC) problem is a natural weighted generalization of the classical edgecoloring problem. Formally, in the MEC problem we are given a graph G = (V,E) with a weight function w : E → N , and we ask for a proper edge-coloring of G, M = {M1,M2, . . . ,Mk}, such that the sum of the weights of the heaviest edges in the color classes, W = ∑k i=1max{w(e)|e ∈ Mi}, is minimized. The MEC problem has been well motivated and studied in the literature. It arises, as a scheduling problem, in switch based communication systems, where messages are transmitted from senders to receivers in a single-hop through direct connections established by an underlying network (e.g., SS/TDMA [8], IQ switch architectures [10]). It is also equivalent to the parallel batch scheduling problem with incompatible jobs, i.e., jobs corresponding to adjacent edges of an underlying graph cannot be scheduled in the same batch [7]. The analogous weighted generalization of the classical vertex-coloring problem has been also studied and it is known as Max VertexColoring (MVC) problem [14]. It is known that it is NP-hard to approximate the MEC problem within a factor less than 7/6 even for cubic planar bipartite graphs with edge weights in {1, 2, 3} [3]. A greedy (2 − 1 ∆)approximation algorithm for general graphs of maximum degree ∆ has been proposed in [10]. For bipartite graphs of ∆ = 3, an optimal 7/6-approximation algorithm is known [3]. Moreover, a series of algorithms presented in [6, 13, 2] improve the 2− 1 ∆ ratio for bipartite graphs of small maximum degrees. However, the ratios of these algorithms either exceed 2 [6, 13] or tend asymptotically to 2 [2] as the maximum degree of the input graph increases. The complexity of the MEC problem in trees remains an open question, while a 3/2-approximation algorithm has been presented in [2]. On the other hand, the MEC problem is known to be polynomial for a few special cases including bipartite graphs with bi-valued edge weights [5], chains [9], stars of chains [13] and bounded degree trees [13]. The analogous MVC problem in general graphs is non appoximable within any constant factor, unless P=NP, as a generalization of the classical vertex-coloring problem. For bipartite graphs, an 8/7 inapproximability result [5, 14] has been attained by an algorithm of the same ratio [3, 14]. The complexity of the MVC problem in trees is also open, but for this problem a PTAS has been presented in [14, 6].