On complexity of the acyclic hypergraph sandwich problem a

Given two hypergraphs H and H ′ on a common vertex set, we write H < H ′ if each edge of H is contained in an edge of H ′. Given H < H ′, either find an acyclic hypergraph A between them, H < A < H ′, or claim that there is no such A. This problem is referred to as the Acyclic Hypergraph Sandwich Problem (AHSP) (H,H ′). We show that one can assume without loss of generality that H is a graph. The AHSP (H,H ′) generalizes the concept of treewidth as follows. Let H = G be a graph, |V (G)| = n, and let H ′ = ( n k ) consists of all subsets of V (G) of cardinality k. Then the AHSP is solvable if and only if the treewidth of G is strictly less than k, that is TW (G) ≤ k − 1. Another important special case of the AHSP is H ′ = Hk, that is the edges of H ′ are the unions of all subfamilies of k edges of H. In this case the AHSP generalizes the hypertreewidth of H. It was recently proved [0] that the ASHP is NP-complete already in case H ′ = H3. However, it is known that verifying TW (G) ≤ k− 1 is polynomial when k is bounded. Respectively, the AHSP (H,H ′) is polynomial when H ′ = ( n k ) . Here we extend this result and show that the AHSP can be solved in time, t = n(d+1)(log n+d+1), where n = |V (H)| and d = dimH ′ is the maximum edge size (so-called dimension) of H ′. In particular, t is quasi-polynomial in n whenever d is bounded or polylogarithmic in n. Hence, in this case the AHSP is not NP-complete unless every problem from NP can be solved in quasi-polynomial time. In particular, the AHSP (H,Hk) is quasi-polynomial, t = n(kd+1)(log n+kd+1), whenever both k and dimH are bounded or polylogarithmic in n.