ALGORITHMS FOR A VARIANT OF THE RESOURCE ALLOCATION PROBLEM

Abstract

In this paper we consider the following variant of the resource allocation problem: Maximize Σ^n_<i=1> xi subject to Σ^n_<i=1> f_i (x_i)〓 R, and xi: nonnegative integers, where f_i's are nondecreasing convex functions defined over [0, ∞) and satisfy Σ^n_<i=1> f_i (0) 〓 R and lim___<x_i→∞> f_i (x_i) = ∞. R is a real number. We present three algorithms for this problem. The first one requires O (N*log n + n) time, where N* denotes the optimal objective value. The second one requires O (n^2 (log N^^~)^2) time, where N^^~ denotes a given upper bound of the optimal value. The third one requires O (b (n, R) + n log n) time, where b (n, R) denotes the computational time required to solve the continuous problem obtained from the original one by dropping the integrality condition on x_i's.