Solution independence and self-referential instances

TL;DR

This paper explores the hitting set problem, highlighting solution independence as key to creating self-referential instances and analyzing their complexity implications.

Contribution

It identifies solution independence as crucial for self-referential instances and distinguishes properties of related problems like vertex cover and dominating set.

Findings

Vertex cover lacks solution independence, enabling search space compression.

Dominating set satisfies solution independence, allowing self-referential instance construction.

Self-referential instances are irreducible, requiring near-complete graph processing for solutions.

Abstract

In this paper, we investigate the hitting set problem and demonstrate that solution independence is the crucial property underlying the construction of self-referential instances. As a special case of the hitting set problem, the vertex cover problem lacks the solution independence property. This distinction accounts for its ability to evade exhaustive search, as correlations among candidate solutions can be leveraged to compress the overall search space. In contrast, the dominating set problem on hypergraphs, which is also a special case of the hitting set problem, satisfies the solution independence property, thereby enabling the construction of self-referential instances. Moreover, we prove that these self-referential instances possess an irreducible property, implying that any algorithm for solving such instances must process nearly the entire graph to yield a correct solution.