CFDP 1458

Neighborhood Complexes and Generating Functions

Author(s):

Publication Date: April 2004

Pages: 24

Abstract:

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 Codes: C61

See CFP: 1169