Communication Complexity of Distributed Resource Allocation Optimization

This paper investigates the communication complexity of distributed resource allocation problems. Communication complexity, which has emerged from computer science, measures the minimal number of communicated bits needed to solve some classes of problems regardless of the used algorithm, when the communication is lossless. We introduce two complexity measures for finding primal/dual saddle point solutions of resource allocation problems. The first measure is -complexity, which quantifies the minimal number of bits needed to find an -accurate solution. The second measure is b-complexity, which quantifies the best possible solution accuracy that can be achieved from communicating b bits. We find the exact and b-complexity of a class of resource allocation problems where a single supplier allocates resources to multiple users. For both the primal and dual problems, the -complexity grows proportionally to log2(1/ ) and the b-complexity proportionally to 1/2 . We also introduce a variant of the and b-complexity measures where only algorithms that ensure primal feasibility of the iterates are allowed. Such algorithms are often desirable because overuse of the resources can overload the respective systems, e.g., by causing blackouts in power systems. We provide upper and lower bounds on the convergence rate of these primal feasible complexity measures. In particular, we show that the b-complexity cannot converge at a faster rate than O(1/b). Therefore, the results demonstrate a trade-off between fast convergence and primal feasibility. We illustrate the result by numerical studies.