Two exact algorithms for the generalized assignment problem

TL;DR

This paper introduces two exact algorithms for the generalized assignment problem, one with O(n^4) time complexity and another with O(n^3) for a special case, improving solution efficiency.

Contribution

It presents a novel O(n^4) algorithm for the generalized assignment problem and an O(n^3) algorithm for the limited-capacity case, using the Hungarian algorithm.

Findings

O(n^4) algorithm for the general problem

O(n^3) algorithm for the limited-capacity case

Efficient solutions using Hungarian algorithm

Abstract

Let A={a_1,a_2,...,a_s} and B={b_1,b_2,...,b_t} be two sets of objects with s+r=n, the generalized assignment problem assigns each element a_i in A to at least alpha_i and at most alpha '_i elements in B, and each element b_j in B to at least beta_j and at most beta '_j elements in A for all 1 <= i <= s and 1 <= j <= t. In this paper, we present an O(n^4) time and O(n) space algorithm for this problem using the well known Hungarian algorithm. We also present an O(n^3) algorithm for a special case of the generalized assignment, called the limited-capacity assignment problem, where alpha_i,beta_j=1 for all i,j.