Computing a many-to-many matching with demands and capacities between two sets using the Hungarian algorithm

TL;DR

This paper introduces an algorithm that extends the Hungarian algorithm to efficiently compute a minimum-cost many-to-many matching with demands and capacities between two sets.

Contribution

It adapts the Hungarian algorithm to handle complex constraints of demands and capacities in many-to-many matchings.

Findings

Successfully computes minimum-cost matchings with demands and capacities.

Extends the Hungarian algorithm to a new class of matching problems.

Provides an efficient solution for complex matching constraints.

Abstract

Given two sets A={a_1,a_2,...,a_s} and {b_1,b_2,...,b_t}, a many-to-many matching with demands and capacities (MMDC) between A and B matches 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 algorithm for finding a minimum-cost MMDC between A and B using the well-known Hungarian algorithm.