On the Computational Complexity of Multi-Objective Ordinal Unconstrained Combinatorial Optimization

TL;DR

This paper investigates the computational complexity of multi-objective unconstrained combinatorial optimization problems, identifying specific cases with up to two ordinal and one real-valued objectives that are solvable in polynomial time.

Contribution

The paper characterizes conditions under which MUCO problems are tractable and provides polynomial-time algorithms for these special cases.

Findings

MUCO problems with up to two ordinal and one real-valued objectives are polynomial-time solvable.

Complete nondominated sets can be computed efficiently for these cases.

Greedy algorithms improve efficiency for problems with one ordinal and another ordinal or real-valued objective.

Abstract

Multi-objective unconstrained combinatorial optimization problems (MUCO) are in general hard to solve, i.e., the corresponding decision problem is NP-hard and the outcome set is intractable. In this paper we explore special cases of MUCO problems that are actually easy, i.e., solvable in polynomial time. More precisely, we show that MUCO problems with up to two ordinal objective functions plus one real-valued objective function are tractable, and that their complete nondominated set can be computed in polynomial time. For MUCO problems with one ordinal and a second ordinal or real-valued objective function we present an even more efficient algorithm that applies a greedy strategy multiple times.