Approximation Algorithms for Generalized Assignment Problems

We consider a class of max-profit scheduling problems that occur naturally in many different applications, all involving assignment of jobs to multiple resources under a set of constraints. In the Max-Profit Generalized Assignment Problem (Max-GAP), we are given a set J of m bins (knapsacks), and a set I of n items. Each bin j ∈ J has capacity c(j). Each item i ∈ I has in bin j size `(i, j) and profit p(i, j). The objective is to find a maximum profit feasible assignment. The problem admits a 1/2approximation algorithm. Our main result is a (1− 1/e)-approximation algorithm for Max-GAP with fixed profits when each item i has a fixed profit p(i, j) = p(i) in every bin j. A particular case of Max-GAP with fixed profits is the Multiple Knapsack with Assignment Restrictions (MKAR) problem, where each bin j ∈ J has capacity c(j) and a specified set I(j) of items that can be assigned to it, and each item i has size `(i) and profit p(i). We show that this version is APX-hard, and give a fast 1/2-approximation algorithm.