A randomized polynomial-time algorithm for the Spanning Hypertree Problem on 3-uniform hypergraphs

Consider the problem of determining whether there exists a spanning hypertree in a given k-uniform hypergraph. This problem is trivially in P for k=2, and is NP-complete for k>= 4, whereas for k=3, there exists a polynomial-time algorithm based on Lovasz’ theory of polymatroid matching. Here we give a completely different, randomized polynomial-time algorithm in the case k=3. The main ingredients are a Pfaffian formula by Vaintrob and one of the authors (G.M.) for a polynomial that enumerates spanning hypertrees with some signs, and a lemma on the number of roots of polynomials over a finite field.

💡 Research Summary

The paper addresses the decision problem of whether a given 3‑uniform hypergraph contains a spanning hypertree (k‑SHT for k = 3). While the problem is trivial for ordinary graphs (k = 2) and NP‑complete for k ≥ 4, the case k = 3 is known to be solvable in polynomial time via Lovász’s polymatroid matching algorithm. The authors present a completely different, randomized polynomial‑time algorithm that belongs to the complexity class RP.

The central mathematical tool is a Pfaffian representation of an alternating‑sign multivariate generating polynomial for spanning hypertrees, originally proved by Vaintrob and Masbaum. For a hypergraph G with vertex set V of size N = 2n + 1 (where n is the number of hyperedges in any spanning hypertree), define a completely antisymmetric N × N matrix Λ whose off‑diagonal entries are \