Max Edge Coloring of Trees

In the standard edge coloring problem we ask for a partition S = {M 1 , M 2 , . . . , M s } of the edge set of G into color classes (matchings) such that s is minimized. In this parer we study the following generalization of the standard edge coloring problem which arises in the domain of optical communication systems (see for example [4]): A positive integer weight is associated with each edge of G and we now ask for a partition S = {M 1 , M 2 , . . . , M s } of the edges of G into color classes, each one of weight w i = max{w(e)|e ∈ M i }, such that the sum of the colors' weights W = s i=1 w i is minimized.

The analogous generalization for the standard vertex coloring problem, where weights are associated to the vertices of a graph and the weight of each color class (independent set) equals to the weight of its heaviest vertex, has been also addressed in the literature and it is known as Max (Vertex) Coloring (MVC) problem [8,7]. Respectively to this we refer to our problem as Max Edge Coloring (MEC) problem.

It is known that the MEC problem is strongly NP-hard and 7/6 inapproximable even for cubic planar bipartite graphs with edge weights w(e) ∈ {1, 2, 3} [1]. On the other hand, the MEC problem is known to be polynomial for a few special cases including bipartite graphs with edge weights w(e) ∈ {1, 2} [2], chains [3] (in fact, this algorithm can be also applied for graphs of ∆ = 2), stars of chains and bounded degree trees [5].

Concerning the approximability of the MEC problem, a natural greedy 2-approximation algorithm for general graphs has been proposed by Kesselman and Kogan [4]. The ratio of this algorithm has been slightly improved to 2 -1 ∆ and 2 -2 ∆+2 in [6]. Especially for bipartite graphs of maximum degree ∆ = 3 an algorithm that attains the 7/6 inapproximability bound has been presented in [1]. For bipartite graphs have been also presented algorithms improving the best known 2 -2 ∆+2 approximation ratio for general graphs. In fact, algorithms presented in [3] and [5] achieve better ratios for bipartite graphs of ∆ ≤ 7, and ∆ ≤ 12, respectively. Moreover, two algorithms of approximation ratios 2 -2 ∆+1 and

∆ 3 +∆ 2 +∆-1 have been presented in [6]. It is interesting that no algorithm of approximation ratio 2 -δ, for any small constant δ > 0, is known for the MEC problem on bipartite graphs or even on trees. Recall that the MEC problem on bipartite graphs is 7/6 inapproximable and notice that neither the complexity of the MEC problem on trees is known. On the other hand, for the MVC problem this gap is closed for bipartite graphs and it is very narrow for trees. In fact, an algorithm which matches the inapproximability bound of 8/7 is known for bipartite graphs [2,1,7] while a PTAS is known for trees [7]. However, the complexity of the MVC problem on trees remains also unknown. In this note we decrease this gap for the MEC problem on trees by presenting a 3/2-approximation algorithm.

We consider the MEC problem on a weighted tree T = (V, E). By d(v) we denote the degree of vertex v ∈ V and by ∆ the maximum degree of T . By S * = {M * 1 , M * 2 , . . . , M * s * } we denote an optimal solution to the MEC problem of weight OP T = w * 1 + w * 2 + . . . + w * s * . For each vertex u ∈ V , we denote by E u : e u 1 , e u 2 , . . . , e u d(u) an ordering of its adjacent edges in non increasing weights, i.e. w(e u 1 ) ≥ w(e u 2 ) ≥ . . . ≥ w(e u d(u) ). Furthermore, we define y i , 1 ≤ i ≤ ∆, to be the weight of the heaviest edge between those ranked i in each ordering E u , u ∈ V , i.e.

Proof: Let e = (u, v) be the heaviest edge with rank equal to i i.e., y i = w(e). For at least one of the endpoints of e, assume w.l.o.g. for u, it holds that e is ranked i in E u , that is y i = w(e u i ). Therefore, there exist i edges adjacent to vertex v of weight at least y i . These i edges belong in i different matchings in an optimal solution, since they share vertex u a common endpoint. Thus, the i-th matching in an optimal solution is of weight at least y i .

In [4], Kesselman and Kogan present the most interesting and general result we have for the MEC problem. This is a greedy 2-approximation algorithm for general graphs, to which we refer as Algorithm KK. A slightly better analysis of this algorithm presented in [6] leads to the following approximation ratio which also matches exactly the ratio of the tightness counterexample given in [4].

Lemma 1 [6] Algorithm KK achieves a tight approximation ratio of 2 -

In this section, we first present a (1 +

w * 1 -w * ∆ OP T )-approximation algorithm for the MEC problem on trees.

Our algorithm roots the tree in an arbitrary vertex r and constructs a solution as following: For each vertex v ∈ V , consider the edges to the children of v in non increasing order and insert them into the first matching they fit.

Algorithm 1 1. Root the tree on an arbitrary vertex r; 2. For each vertex v in pre-order do 3.

Sort the children edges of v in non-increasing order,

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