Complexity of the Multiobjective Spanner Problem

TL;DR

This paper investigates the computational complexity of the Multiobjective Spanner problem, revealing its intractability and hardness results for specific graph classes, which impacts its application in infrastructure planning.

Contribution

It provides the first complexity analysis of the Multiobjective Spanner problem, establishing intractability and hardness results for various graph classes and problem variants.

Findings

MSp is intractable for degree-3 bounded outerplanar instances.

Computing the non-dominated set is BUCO-hard.

No output-polynomial algorithms exist under P != NP for certain cases.

Abstract

In this paper, we take an in-depth look at the complexity of a hitherto unexplored Multiobjective Spanner (MSp) problem. The MSp is a multiobjective generalization of the well-studied Minimum t-Spanner problem. This multiobjective approach allows us to find solutions that offer a viable compromise between cost and utility. Thus, the MSp can be a powerful modeling tool when it comes to the planning of, e.g., infrastructure. We show that for degree-3 bounded outerplanar instances the MSp is intractable and computing the non-dominated set is BUCO-hard. Additionally, we prove that if P != NP, neither the non-dominated set nor the set of extreme points can be computed in output-polynomial time, for instances with unit costs and arbitrary graphs. Furthermore, we consider the directed versions of the cases above.