Biobjective optimization with M-convex functions

TL;DR

This paper introduces polynomial-time algorithms for biobjective optimization problems involving M-convex functions, combining multiobjective optimization with discrete convex analysis for the first time.

Contribution

It demonstrates that the entire Pareto set can be efficiently computed for problems with M-convex functions and linear functions, and provides specialized algorithms for M-convex cases.

Findings

Entire Pareto optimal set can be obtained in polynomial time.

More efficient algorithms are available for M-convex functions.

Polynomial-time methods are developed for lexicographic biobjective optimization.

Abstract

In this paper, we deal with two ingredients that, as far as we know, have not been combined until now: multiobjective optimization and discrete convex analysis. First, we show that the entire Pareto optimal value set can be obtained in polynomial time for biobjective optimization problems with discrete convex functions, in particular, involving an M$^\natural$-convex function and a linear function with binary coefficients. We also observe that a more efficient algorithm can be obtained in the special case where the M$^\natural$-convex function is M-convex. Additionally, we present a polynomial-time method for biobjective optimization problems that combine M$^\natural$-convex function minimization with lexicographic optimization.