Exploiting Structure in the Bottleneck Assignment Problem

TL;DR

This paper presents a novel approach to solving the bottleneck assignment problem by exploiting its structure to decompose and merge subproblems, leading to more efficient solutions demonstrated through a case study.

Contribution

It introduces a method to decompose the BAP into subproblems, merge solutions, and provides conditions for exact solutions, along with a new algorithm leveraging this structure.

Findings

The proposed method reduces computational complexity for certain BAP instances.

Conditions are identified where subproblem solutions yield exact solutions.

Numerical case study demonstrates the effectiveness of the approach.

Abstract

An assignment problem arises when there exists a set of tasks that must be allocated to a set of agents. The bottleneck assignment problem (BAP) has the objective of minimising the most costly allocation of a task to an agent. Under certain conditions the structure of the BAP can be exploited such that subgroups of tasks are assigned separately with lower complexity and then merged to form a combined assignment. In particular, we discuss merging the assignments from two separate BAPs and use the solution of the subproblems to bound the solution of the combined problem. We also provide conditions for cases where the solution of the subproblems produces an exact solution to the BAP over the combined problem. We then introduce a particular algorithm for solving the BAP that takes advantage of this insight. The methods are demonstrated in a numerical case study.