An adjacency-based algorithm for computing all extreme-supported non-dominated points of a bi-objective combinatorial optimisation problem

TL;DR

This paper introduces a neighborhood-based algorithm for finding all extreme-supported non-dominated points in bi-objective combinatorial problems, offering an alternative to dichotomic search with proven efficiency under certain adjacency conditions.

Contribution

It proposes a generic adjacency-based algorithm for bi-objective problems and demonstrates its effectiveness and polynomial runtime on the bi-objective minimum weight basis problem.

Findings

The algorithm guarantees success under specific adjacency conditions.

It has polynomial runtime for the bi-objective minimum weight basis problem.

A specialized version for matroids swaps directly between extreme-supported points.

Abstract

Generally, multi-objective optimisation problems are solved exactly or approximated by solving a series of scalarisations, for example by dichotomic search. In this paper, we take a different approach and attempt to compute the set of all extreme-supported non-dominated points of a bi-objective combinatorial optimisation problem by using a neighbourhood-based approach. Whether or not this works depends on the definition of adjacency and we provide sufficient conditions that guarantee its success. The resulting generic algorithm is an alternative to dichotomic search in our setting. We then apply our generic algorithm to a specific example: the bi-objective minimum weight basis problem, in which we are given a matroid and want to find bases of minimum weight. We use the natural definition of adjacency, in which two bases are adjacent if they differ in exactly one element. Since this satisfies our sufficient condition on the adjacency relation, our generic algorithm works in this case and we analyse its running time, showing that it is polynomial. By tailoring this algorithm specifically to matroids, we obtain one that is faster but no longer transitions between adjacent solutions, instead swapping directly from one extreme-supported point to the next in a combinatorial fashion.