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

The Minimum Dominating Set problem is NP-hard, and the best known polynomial-time approximation factor is O(ln n), which is provably tight unless P = NP. We present a polynomial-time algorithm that reduces an arbitrary input graph to a planar kernel through forced-vertex extraction, pendant elimination, and greedy planarisation, and then applies Baker’s PTAS to that kernel. The algorithm runs in O(mn + m log m) time — in particular O(n log n) on sparse graphs — and is provably within twice the optimum whenever the reduction is tight. We give a structural witness mapping that injects the post-pruning forced-boundary set into the rest of the planar kernel, narrowing the unresolved gap in the analysis to a single inequality, |F| ≥ 2|F_R^pruned|. Should that inequality hold universally, a 2-approximation would follow and would imply P = NP. We complement the theory with an experimental study on thirteen DIMACS benchmark graphs: in every case the algorithm finishes in well under five minutes and returns a dominating set whose size is at most 1.80× the ILP optimum, with an average ratio of 1.42. An open-source implementation is provided as the Furones package (v0.2.6).