Rate Region of Scheduling a Wireless Network with Discrete Propagation Delays

We study wireless networks where signal propagation delays are multiples of a time interval. Such a network can be modelled as a weighted hypergraph. The link scheduling problem of such a wireless network is closely related to the independent sets of the periodic hypergraph induced by the weighted hypergraph. As the periodic graph has infinitely many vertices, existing characterizations of graph independent sets cannot be applied to study link scheduling efficiently. To characterize the rate region of link scheduling, a directed graph of finite size called scheduling graph is derived to capture a certain conditional independence property of link scheduling over time. A collision-free schedule is equivalent to a path in the scheduling graph, and hence the rate region is equivalent to the convex hull of the rate vectors associated with the cycles of the scheduling graph. With the maximum independent set problem as a special case, calculating the whole rate region is NP hard and also hard to approximate. We derive two algorithms that benefit from a partial order on the paths in the scheduling graph, and can potentially find schedules that are not dominated by the existing cycle enumerating algorithms running in a given time. The first algorithm calculates the rate region incrementally in the cycle lengths so that a subset of the rate region corresponding to short cycles can be obtained efficiently. The second algorithm enumerates cycles associated with a maximal subgraph of the scheduling graph. In addition to scheduling a wireless network, the independent sets of periodic hypergraphs also find applications in some operational research problems.