Scheduling Wireless Links in the Physical Interference Model by Fractional Edge Coloring

Abstract

We consider the problem of scheduling the links of wireless mesh networks for capacity maximization in the physical interference model. We represent such a network by an undirected graph $G$, with vertices standing for network nodes and edges for links. We define network capacity to be $1/\chi'^*_\mathrm{phys}(G)$, where $\chi'^*_\mathrm{phys}(G)$ is a novel edge-chromatic indicator of $G$, one that modifies the notion of $G$'s fractional chromatic index. This index asks that the edges of $G$ be covered by matchings in a certain optimal way. The new indicator does the same, but requires additionally that the matchings used be all feasible in the sense of the physical interference model. Sometimes the resulting optimal covering of $G$'s edge set by feasible matchings is simply a partition of the edge set. In such cases, the index $\chi'^*_\mathrm{phys}(G)$ becomes the particular case that we denote by $\chi'_\mathrm{phys}(G)$, a similar modification of $G$'s well-known chromatic index. We formulate the exact computation of $\chi'^*_\mathrm{phys}(G)$ as a linear programming problem, which we solve for an extensive collection of random geometric graphs used to instantiate networks in the physical interference model. We have found that, depending on node density (number of nodes per unit deployment area), often $G$ is such that $\chi'^*_\mathrm{phys}(G)<\chi'_\mathrm{phys}(G)$. This bespeaks the possibility of increased network capacity by virtue of simply defining it so that edges are colored in the fractional, rather than the integer, sense.