A ( 1 − 1 / e )-approximation algorithm for the maximum generalized assignment problem with fixed profits

The Max-Profit Generalized Assignment Problem (Max-GAP) is: given sets J of bins and I of items, where each j ∈ J has capacity c(j) and each i ∈ I has in bin j size `(i, j) and profit p(i, j), find a maximum profit feasible assignment. The problem admits a 1/2-approximation algorithm. Our main result is a (1− 1/e)-approximation algorithm for Max-GAP with fixed profits when each i ∈ I has a fixed profit p(i, j) = p(i). We also give a fast 1/2-approximation algorithm for the Multiple Knapsack with Assignment Restrictions (MKAR) problem. Key-words. Generalized assignment, Approximation algorithm.