Soft Edge Coloring

Abstract

We consider the following channel assignment problem arising in wireless networks. We are given a graph G = (V, E), and the number of wireless cards C _v for all v, which limit the number of colors that edges incident to v can use. We also have the total number of channels C _G available in the network. For a pair of edges incident to a vertex, they are said to be conflicting if the colors assigned to them are the same. Our goal is to color edges (assign channels) so that the number of conflicts is minimized. We first consider the homogeneous network where C _v  = k and C _G  ≥ C _v for all nodes v. The problem is NP-hard by a reduction from Edge coloring and we present two combinatorial algorithms for this case. The first algorithm is a distributed greedy method, which gives a solution with at most $(1 - \frac{1}{k})|E|$ more conflicts than the optimal solution. We also present an algorithm yielding at most |V| more conflicts than the optimal solution. The algorithm generalizes Vizing’s algorithm in the sense that it gives the same result as Vizing’s algorithm when k = Δ + 1. Moreover, we show that this approximation result is best possible unless P = NP. For the case where C _v  = 1 or k, we show that the problem is NP-hard even when C _v  = 1 or 2, and C _G  = 2, and present two algorithms. The first algorithm is completely combinatorial and produces a solution with at most $(2-\frac{1}{k}) OPT + (1 - \frac{1}{k}) |E|$ conflicts. We also develop an SDP-based algorithm, producing a solution with at most 1.122 OPT + 0.122 |E| conflicts for k = 2, and $(2-\Theta(\frac{\ln k}{k})) OPT + (1 - \Theta(\frac{\ln k}{k}))|E|$ conflicts in general.