Enumerating minimal solution sets for metric graph problems

TL;DR

This paper explores the enumeration of minimal solution sets for metric graph problems, establishing connections to hypergraph transversal enumeration and providing new insights into their computational complexity.

Contribution

It introduces the enumeration of minimal resolving and geodetic sets as equivalent to hypergraph transversal enumeration, advancing understanding of their complexity and properties.

Findings

Enumerating minimal resolving sets in graphs is equivalent to Trans-Enum.

Enumerating minimal geodetic sets in split graphs is equivalent to Trans-Enum.

Minimal strong resolving sets can be enumerated with polynomial delay.

Abstract

Problems from metric graph theory like Metric Dimension, Geodetic Set, and Strong Metric Dimension have recently had a strong impact in parameterized complexity by being the first known problems in NP to admit double-exponential lower bounds in the treewidth, and even in the vertex cover number for the latter, assuming the Exponential Time Hypothesis. We initiate the study of enumerating minimal solution sets for these problems and show that they are also of great interest in enumeration. Specifically, we show that enumerating minimal resolving sets in graphs and minimal geodetic sets in split graphs are equivalent to enumerating minimal transversals in hypergraphs (denoted Trans-Enum), whose solvability in total-polynomial time is one of the most important open problems in algorithmic enumeration. This provides two new natural examples to a question that emerged in recent works: for which vertex (or edge) set graph property $\Pi$ is the enumeration of minimal (or maximal) subsets satisfying $\Pi$ equivalent to Trans-Enum? As very few properties are known to fit within this context -- namely, those related to minimal domination -- our results make significant progress in characterizing such properties, and provide new angles to approach Trans-Enum. In contrast, we observe that minimal strong resolving sets can be enumerated with polynomial delay. Additionally, we consider cases where our reductions do not apply, namely graphs with no long induced paths, and show both positive and negative results related to the enumeration and extension of partial solutions.