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

TL;DR

This paper studies multi-objective unconstrained combinatorial optimization (MUCO) and shows a tractable boundary when there are at most two ordinal objectives and one real-valued objective. The authors formulate a series of epsilon-constraint scalarizations whose constraint matrices are totally unimodular, allowing polynomial-time solutions via linear programming relaxations; they further bound the number of potentially nondominated outcomes by $|U^e|=O(n^{\tilde{K}+\hat{K}-1})$, enabling a complete polynomial-time computation of the nondominated set for fixed ordinal categories $\tilde{K},\hat{K}$. For the bi-objective case, they develop a greedy approach grounded in a partition matroid that yields optimal solutions efficiently, with complexity $O(n^{2\tilde{K}})$. Overall, the work identifies a meaningful tractable subclass of MUCO problems and provides practical algorithms, while suggesting future work on the structure and connectivity of the nondominated set and extensions to more objectives.

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.

Theorems & Definitions (17)

• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Corollary 1
• proof
• Corollary 2
• proof