Self-referential instances of the dominating set problem are irreducible
For theoretical computer science, it establishes that the hardness of the dominating set problem in random graphs is due to global self-referential structure, not local features.
The paper shows that for random graphs G(n,p) with a specific edge probability, no algorithm inspecting a subgraph of size n^c (c<1) can determine if the graph has a dominating set of size ln n, proving the problem is irreducible and requires exhaustive search.
We study the algorithmic decidability of the domination number in the Erdos-Renyi random graph model $G(n,p)$. We show that for a carefully chosen edge probability $p=p(n)$, the domination problem exhibits a strong irreducible property. Specifically, for any constant $0<c<1$, no algorithm that inspects only an induced subgraph of order at most $n^c$ can determine whether $G(n,p)$ contains a dominating set of size $k=\ln n$. We demonstrate that the existence of such a dominating set can be flipped by a local symmetry mapping that alters only a constant number of edges, thereby producing indistinguishable random graph instances which require exhaustive search. These results demonstrate that the extreme hardness of the dominating set problem in random graphs cannot be attributed to local structure, but instead arises from the self-referential nature and near-independence structure of the entire solution space.