Self-referential instances of the dominating set problem are irreducible

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.

💡 Research Summary

The paper investigates the algorithmic decidability of the domination number in the Erdős‑Rényi random graph model G(n,p). The authors introduce a new notion of “reducibility”: a domination‑set instance is reducible if there exists an induced subgraph H of size at most n^c (for some constant 0<c<1) such that H contains a dominating set of size k if and only if the whole graph G does. Otherwise the instance is called irreducible. The main theorem asserts that, for a carefully tuned edge probability p=p(n), the domination‑set problem with target size k=ln n is irreducible for every constant c∈(0,1). In other words, no algorithm that inspects only a sublinear‑size subgraph can correctly decide whether G contains a dominating set of size k.

The proof proceeds in two stages. First, the authors select p≈1−e^{-(1−ε_n)ln 2 n}/ln n with ε_n≈(ln ln n)/ln n. Under this choice the expected number X of dominating sets of size k satisfies E