Submodularity and Its Applications in Wireless Communications

Abstract

This monograph studies the submodularity in wireless communications and how to use it to enhance or improve the design of the optimization algorithms. The work is done in three different systems.

In a cross-layer adaptive modulation problem, we prove the submodularity of the dynamic programming (DP), which contributes to the monotonicity of the optimal transmission policy. The monotonicity is utilized in a policy iteration algorithm to relieve the curse of dimensionality of DP. In addition, we show that the monotonic optimal policy can be determined by a multivariate minimization problem, which can be solved by a discrete simultaneous perturbation stochastic approximation (DSPSA) algorithm. We show that the DSPSA is able to converge to the optimal policy in real time.

For the adaptive modulation problem in a network-coded two-way relay channel, a two-player game model is proposed. We prove the supermodularity of this game, which ensures the existence of pure strategy Nash equilibria (PSNEs). We apply the Cournot tatonnement and show that it converges to the extremal, the largest and smallest, PSNEs within a finite number of iterations. We derive the sufficient conditions for the extremal PSNEs to be symmetric and monotonic in the channel signal-to-noise (SNR) ratio.

Based on the submodularity of the entropy function, we study the communication for omniscience (CO) problem: how to let all users obtain all the information in a multiple random source via communications. In particular, we consider the minimum sum-rate problem: how to attain omniscience by the minimum total number of communications. The results cover both asymptotic and non-asymptotic models where the transmission rates are real and integral, respectively. We reveal the submodularity of the minimum sum-rate problem and propose polynomial time algorithms for solving it. We discuss the significance and applications of the fundamental partition, the one that gives rise to the minimum sum-rate in the asymptotic model. We also show how to achieve the omniscience in a successive manner.