Wireless Scheduling with Power Control

We consider the scheduling of arbitrary wireless links in the physical model of interference to minimize the time for satisfying all requests. We study here the combined problem of scheduling and power control, where we seek both an assignment of power settings and a partition of the links so that each set satisfies the signal-to-interference-plus-noise (SINR) constraints. We give an algorithm that attains an approximation ratio of O(log n loglog Delta), where n is the number of links and Delta is the ratio between the longest and the shortest linklength. Under the natural assumption that lengths are represented in binary, this gives the first polylog(n)-approximation. The algorithm has the desirable property of using an oblivious power assignment, where the power assigned to a sender depends only on the length of the link. We give evidence that this dependence on Delta is unavoidable, showing that any reasonably-behaving oblivious power assignment results in a Omega(loglog Delta)-approximation. We also give a simple online algorithm that yields a O(log Delta)-approximation, by a reduction to the coloring of unit-disc graphs. These results hold also for the (weighted) capacity problem of finding a maximum (weighted) subset of links that can be scheduled in a single time slot. In addition, we obtain improved approximation for a bidirectional variant of the scheduling problem, give partial answers to questions about the utility of graphs for modeling physical interference, and generalize the setting from the standard 2-dimensional Euclidean plane to doubling metrics.