An Approximate Solution to the Minimum Dominating Set Problem: The Furones Algorithm

We present Furones, a linear-time candidate-comparison algorithm for the Minimum Dominating Set (MDS) problem on undirected graphs. The algorithm (version v0.3.8) applies a TSCC-style pendant cascade, solves the reduced instance by a Baker-style routine used only as a validated candidate generator, and compares the lifted candidate against several original-graph candidates: closed-degree coverage, witness sweeps, order ownership, seed completion, a Salvador-style auxiliary, a max-cut double-cover auxiliary, and reverse-delete scans. We prove two unconditional guarantees. First, every normal return is a valid dominating set. Second, because the portfolio contains the dynamic greedy maximum-coverage dominator, every returned set D satisfies |D| ≤ H(∆ + 1) γ(G) ≤ (1 + ln(∆ + 1)) γ(G), where ∆ is the maximum degree and H(k) is the k-th harmonic number; this is a constant factor on bounded-degree graphs and sub-logarithmic in n for sub-polynomial degree. We also record the near-threshold ratio hypothesis that Furones meets ratio max{4, ln n} on all graphs; proving it would imply P = NP. The hypothesis is already proved for all graphs with ∆ ≤ √n/e. Exact benchmarks on 11,000 small instances with exhaustive optimum certificates record zero violations of the conjectured bound.