Abstract
In this paper we introduce and study a new problem named \emph{min-max edge $q$-coloring} which is motivated by applications in wireless mesh networks. The input of the problem consists of an undirected graph and an integer $q$. The goal is to color the edges of the graph with as many colors as possible such that: (a) any vertex is incident to at most $q$ different colors, and (b) the maximum size of a color group (i.e. set of edges identically colored) is minimized. We show the following results: 1. Min-max edge $q$-coloring is NP-hard, for any $q \ge 2$. 2. A polynomial time exact algorithm for min-max edge $q$-coloring on trees. 3. Exact formulas of the optimal solution for cliques and almost tight bounds for bicliques and hypergraphs. 4. A non-trivial lower bound of the optimal solution with respect to the average degree of the graph. 5. An approximation algorithm for planar graphs.
Highlights
Backbone connectivity in networks of various sizes has been built using wired infrastructure
Multiple radio channels in the network means, that at least some of the nodes need to handle more than one channel at a time
A couple of such multi-NIC architectures have been proposed by Raniwala et al [13,12]. Their simulation and testbed experiments show a promising improvement with only two NICs per node, compared to a single-channel wireless mesh networks (WMNs)
Summary
Backbone connectivity in networks of various sizes has been built using wired infrastructure. A couple of such multi-NIC (network interface card) architectures have been proposed by Raniwala et al [13,12] Their simulation and testbed experiments show a promising improvement with only two NICs per node, compared to a single-channel WMN. For a hypergraph Qn we give a lower bound which is tight for even n, and almost tight for odd n These classes of graphs have a very simple structure, finding lower bounds is much more difficult than in the case of the max edge q-coloring problem. For the min-max edge q-coloring problem, a trivial lower bound is half of the maximum degree.
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