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 $O(\log n \cdot \log\log \Delta)$, where $n$ is the number of links and $\Delta$ is the ratio between the longest and the shortest link length. Under the natural assumption that lengths are represented in binary, this gives the first approximation ratio that is polylogarithmic in the size of the input. 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 $\Omega(\log\log \Delta)$-approximation. 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. Finally, we explore the utility of graph models in capturing wireless interference.Comment: Revised full versio

Self-similar solutions to the mean curvature flow in the Minkowski plane R1,1\mathbf R^{1,1}

We introduce the mean curvature flow of curves in the Minkowski plane $\mathbf R^{1,1}$ and give a classification of all the self-similar solutions. In addition, we describe five other exact solutions to the flow.Comment: 31 pages, 38 figures. Two exact solutions added from previous versio

Wireless Link Capacity under Shadowing and Fading

We consider the following basic link capacity (a.k.a., one-shot scheduling) problem in wireless networks: Given a set of communication links, find a maximum subset of links that can successfully transmit simultaneously. Good performance guarantees are known only for deterministic models, such as the physical model with geometric (log-distance) pathloss. We treat this problem under stochastic shadowing under general distributions, bound the effects of shadowing on optimal capacity, and derive constant approximation algorithms. We also consider temporal fading under Rayleigh distribution, and show that it affects non-fading solutions only by a constant-factor. These can be combined into a constant approximation link capacity algorithm under both time-invariant shadowing and temporal fading.Comment: 20 pages. In Mobihoc'1