COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 1458

Neighborhood Complexes and Generating Functions

Herbert E. Scarf and Kevin M. Woods

April 2004

Given a₁, a₂,...,a_n in Z^d, we examine the set, G, of all nonnegative integer combinations of these a_i. In particular, we examine the generating function f(z) = Sum_{{b in G}}z^b. We prove that one can write this generating function as a rational function using the neighborhood complex (sometimes called the complex of maximal lattice-free bodies or the Scarf complex) on a particular lattice in Zⁿ. In the generic case, this follows from algebraic results of D. Bayer and B. Sturmfels. Here we prove it geometrically in all cases, and we examine a generalization involving the neighborhood complex on an arbitrary lattice.

Keywords: Integer programming, Complex of maximal lattice free bodies, Generating functions

JEL Classification: C61