On partitioning of hypergraphs

We study Edge-Isoperimetric Problems (EIP) for hypergraphs and extend some technique in this area from graphs to hypergraphs. In particular, we establish some new results on a relationship between the EIP and some extremal poset problems, and apply them to obtain an exact solution of the EIP for certain hypergraph families. We also show how to solve the EIP on hypergraphs in some cases when the link to posets does not work. Another outcome of our results is a new series of hypergraphs admitting nested solutions in the EIP. 1 Problem statement and motivation Let H = {VH , EH} be a hypergraph where VH is the the vertex set and EH ⊆ 2H is the set of hyperedges. For A ⊆ VH denote A = VH \ A and θH(A) = {e ∈ EH | e ∩ A 6= ∅ and e ∩ A 6= ∅} θH(m) = min A⊆VH |A|=m |θH(A)|. In other words, θH(A) is the set of hyperedges connecting A with its complement in VH , or, what is commonly said, θH(A) is the hyperedge-cut. The Edge-Isoperimetric Problem (EIP) on H consists of finding for a given m, 1 ≤ m ≤ |VH |, a set A ⊆ VG such that |A| = m and |θH(A)| = θH(m). This problem is known to be NP-complete even for graphs. We mostly will be dealing with another version of the EIP, which also makes sense for practical applications. Instead of minimizing the number of broken connections we max-