Abstract. Defant, Li, Propp, and Young recently resolved two enumerative conjectures of Propp concerning the tilings of regions in the hexagonal grid called benzels using two types of prototiles called stones and bones (with varying constraints on allowed orientations of the tiles). Their primary tool, a bijection called compression that converts certain [Formula: see text]-ribbon tilings to [Formula: see text]-ribbon tilings, allowed them to reduce their problems to the enumeration of dimers (i.e., perfect matchings) of certain graphs. We present a generalized version of compression that no longer relies on the perspective of partitions and skew shapes. Using this strengthened tool, we resolve three more of Propp’s conjectures and recast several others as problems about perfect matchings.

A natural product on integer programming problems with nonnegative coefficients is defined. Hypergraph covering problems are a special case of such integer programs, and the product defined is a generalization of the usual hypergraph product. The main theorem of this paper gives a sufficient condition under which the solution to the nth power of an integer program is asymptotically as good as the solution to the same nth power when the variables are not necessarily integral but may be arbitrary nonnegative real numbers.