CFDP 1458

Neighborhood Complexes and Generating Functions

Author(s): 

Publication Date: April 2004

Pages: 24

Abstract: 

Given a1, a2,…,an in Zd, we examine the set, G, of all nonnegative integer combinations of these ai. In particular, we examine the generating function f(z) = Sum{b in G}zb. 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