Combinatorial Resource Allocation Using Submodularity of Waterfilling

Abstract

We show that the maximum mutual information (capacity) of parallel Gaussian channels obtained by the optimal water-filling algorithm for power allocation under a sum-power constraint is submodular. For a given power allocation, mutual information is known to be submodular. However, establishing the submodularity of the capacity, which additionally involves maximization of the mutual information over the power allocation, is challenging. Capacity of parallel Gaussian channels is equivalent to the maximum log-utility function used in resource allocation problems. Using this correspondence and the submodularity of the capacity, we find provable guarantees on multiple combinatorial resource allocation problems in wireless networks. In particular, we show that greedy algorithms give a 2-approximation for uplink OFDMA power and subcarrier allocation, FDMA capacity, downlink base-station association, with honest as well as strategic users who may or may not report their channel gains truthfully.