Solution independence and self-referential instances
This work provides theoretical insight into the structural properties that determine computational hardness for hitting set variants, but is incremental as it builds on known concepts.
The paper identifies solution independence as the key property enabling self-referential instances in the hitting set problem, and shows that while vertex cover lacks this property, dominating set on hypergraphs possesses it, leading to irreducible instances that force any algorithm to process nearly the entire graph.
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.