Complexity and algorithms for graph and hypergraph sandwich problems

The script F sign Hypergraph Sandwich Problem (script F signHSP) is introduced here as follows: Given two hypergraphs H¹ = (X, script E sign¹) and H² = (X, script E sign²) where script E sign¹ = {E₁¹, . . . , E_m¹}, script E sign² = {E₁², . . . , E_m²} and E_i¹ ⊆ E_i² for all 1 ≤ i ≤ m, is there a hypergraph H = (X, script E sign) with script E sign = {E₁, . . . , E_m} such that E_i¹ ⊆ E_i ⊆ E_i² for all 1 ≤ i ≤ m which belongs to a specified hypergraph family script F sign? Hypergraph sandwich problems for several properties studied here occur in a variety of important applications. We prove the NP-completeness of the Interval HSP and the Circular-arc HSP. This corresponds to the problem of deciding whether a partially specified (0,1)-valued matrix can be filled in such that the resulting 0/1 matrix has the consecutive ones property, (resp., circular ones property). The consecutive ones property arises in databases and in DNA physical mapping. Further results shown are a set of conditions relating interval hypergraphs with acyclic hypergraphs. Finally, the k-tree graph sandwich problem is studied. The general problem is shown to be NP-complete and the fixed k version is given a polynomial algorithm. Both problems are based on solutions to the corresponding partial k-tree recognition problems.