A Greedy and Distributable Approach to the Lexicographic Bottleneck Assignment Problem with Conditions on Exactness

TL;DR

This paper introduces a greedy, distributable algorithm for the Lexicographic Bottleneck Assignment Problem that reduces computational complexity and applies to multi-agent systems, demonstrated through a mobile robots case study.

Contribution

It presents a cubic complexity algorithm for the SeqBAP, analyzes conditions for equivalence with LexBAP, and enables distributed solutions in networked environments.

Findings

The algorithm solves SeqBAP with cubic complexity.

Conditions for solution set equivalence are characterized.

Distributed implementation reduces computational load in multi-agent systems.

Abstract

Solving the Lexicographic Bottleneck Assignment Problem (LexBAP) typically relies on centralised computation with order quartic complexity. We consider the Sequential Bottleneck Assignment Problem (SeqBAP), which yields a greedy solution to the LexBAP and discuss the relationship between the SeqBAP, the LexBAP, and the Bottleneck Assignment Problem (BAP). In particular, we reexamine tools used to analyse the structure of the BAP, and apply them to derive an algorithm that solves the SeqBAP with cubic complexity. We show that the set of solutions of the LexBAP is a subset of the solutions of the SeqBAP and analyse the conditions for which the solutions sets are identical. Furthermore, we provide a method to verify the satisfaction of these conditions. In cases where the conditions are satisfied, the proposed algorithm for solving the SeqBAP solves the LexBAP with computation that has lower complexity and can be distributed over a network of computing agents. The applicability of the approach is demonstrated with a case study where mobile robots are assigned to goal locations.