A Decomposition Algorithm for Nested Resource Allocation Problems

Abstract

We propose an exact polynomial algorithm for a resource allocation problem with convex costs and constraints on partial sums of resource consumptions, in the presence of either continuous or integer variables. No assumption of strict convexity or differentiability is needed. The method solves a hierarchy of resource allocation subproblems, whose solutions are used to convert constraints on sums of resources into new bounds for variables at higher levels. The resulting time complexity for the integer problem is O(n log m log(B/n)), and the complexity of obtaining an ∈-approximate solution for the continuous case is O(n log m log(B/∈)), n being the number of variables, m the number of ascending constraints (such that m ≤ n), ∈ a desired precision, and B the total resource. This algorithm matches the best-known complexity when m = n and improves it when log m = o(log n). Extensive experimental analyses are presented with four recent algorithms on various continuous problems issued from theory and practice. The proposed method achieves a better performance than previous algorithms, solving all problems with up to 1 million variables in less than 1 minute on a modern computer.