Complexity and Algorithms for Graph and Hypergraph Sandwich Problems

Abstract.

 The ℱ Hypergraph Sandwich Problem (ℱHSP) is introduced here as follows: Given two hypergraphs H ¹=(X,ℰ¹) and H ²=(X,ℰ²) where ℰ¹={E ₁ ¹,…,E _m ¹}, ℰ²={E ₁ ²,…,E _m ²} and E _i ¹⊆E _i ² for all 1≤i≤m, is there a hypergraph H=(X,ℰ) with ℰ={E ₁,…,E _m } such that E _i ¹⊆E _i ⊆E _i ² for all 1≤i≤m which belongs to a specified hypergraph family ℱ? 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.