Two exact algorithms for the generalized assignment problem

Let A = {a1, a2, . . . , as} and {b1, b2, . . . , bt} be two sets of objects with s + r = n, the generalized assignment problem assigns each element ai ∈ A to at least αi and at most α ′ i elements in B, and each element bj ∈ B to at least βj and at most β ′ j elements in A for all 1 ≤ i ≤ s and 1 ≤ j ≤ t. In this paper, we present an O(n) time and O(n) space algorithm for this problem using the well known Hungarian algorithm. We also present an O(n) algorithm for a special case of the generalized assignment, called the limited-capacity assignment problem, where αi, βj = 1 for all i, j.