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

Sergio Caracciolo; Gregor Masbaum; Alan D. Sokal; Andrea Sportiello

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arXiv · 0812.3593

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

Sergio Caracciolo·Gregor Masbaum·Alan D. Sokal·Andrea Sportiello

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Abstract

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.

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Sergio Caracciolo, Gregor Masbaum, Alan D. Sokal, Andrea Sportiello. 2008-12-18. A randomized polynomial-time algorithm for the Spanning Hypertree Problem on 3-uniform hypergraphs. https://arxiv.org/abs/0812.3593

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