A Fast Distributed Algorithm for α-Fair Packing Problems

Over the past two decades, fair resource allocation problems received considerable attention in avariety of application areas. While polynomial time distributed algorithms have been designed formax-min fair resource allocation, the design of distributed algorithms with convergence guarantees forthe more general α−fair allocations received little attention. In this paper, we study weighted α-fairpacking problems, that is, the problems of maximizing the objective functions∑j wjx1−αj /(1 − α)when α 6= 1 andj wj lnxj when α = 1 over linear constraints Ax ≤ b, x ≥ 0, where wj arepositive weights and A and b are non-negative. We consider the distributed computation modelthat was used for packing linear programs and network utility maximization problems. Under thismodel, we provide a distributed algorithm for general α. The algorithm uses simple local update rulesand is stateless (namely, it allows asynchronous updates, is self-stabilizing, and allows incrementaland local adjustments). It converges to approximate solutions in running times that have an inversepolynomial dependence on the approximation parameter ε. The convergence time has polylogarithmicdependence on the problem size for α 6= 1, and a nearly-linear dependence on the number of variablesfor α = 1. These are the best convergence times known for these problems.