Links' Seminars and Public Events |
Fri, March 9, 2018 10:00 am 11:00 am | Benjamin Bergougnoux : Counting minimal transversals of hypergraphs A transversal of a hypergraph H is a subset of vertices that intersects all the hyper-edges H. The enumeration and the counting of the minimal transversals have a lot of applications in many domains (graph theory, AI, datamining, etc). Counting problems are generally harder than theirs associated decision problems. For example, finding a minimal transversal is doable in polynomial time but counting them is #P-complet (the equivalent of NP-complet for counting problems). We have proved that we can count the minimal transversals of any beta-acyclique hypergraph in polynomial time. Our result is based on a recursive decomposition of the beta-acyclique hypergraph founded by Florent Capelli and by the introduction of a new notion that generalize the minimal transversals. A lot of exciting open questions live in the neighborhood of our result. In particular, our algorithm is able to count the minimum dominating set of a strong-chordal graph. But counting the minimum dominating set is #P-complete on split graphs. Is it the beginning of a complete characterization of the complexity of counting minimal dominating sets in dense graphs ? Salle B21 |