Perfect matching in bipartite hypergraphs subject to a demand graph

Motivated by the problem of assigning plots to tenants, we present a version of the bipartite hypergraph matching problem. This version deals with a hypergraph with a constraint on its hyperedges, defined by a demand graph. We study the complexity of the matching problem for different demand graphs. The matching problem for 3-uniform hypergraphs is polynomially solvable if the set of perfect matchings of the demand graph can be polynomially generated. On the other hand, when the number of disjoint even cycles in the demand graph is Ω (n^1/k) , for some constant k , the matching problem is NP-complete. For non-uniform hypergraphs, we show that the problem is NP-complete, even for very simple demand graphs.