A Yannakakis-type theorem for lifts of affine semigroups

TL;DR

This paper extends Yannakakis' theorem to affine semigroups, establishing a connection between their lifts and nonnegative factorizations, with implications for integer programming and matrix rank analysis.

Contribution

It introduces a new framework for affine semigroups analogous to convex polytopes, including definitions of slack matrices and lifts, and demonstrates their properties and applications.

Findings

Established a Yannakakis-type theorem for affine semigroups.

Defined integer slack matrices and lifts for affine semigroups.

Applied slack matrices to analyze nonnegative integer rank.

Abstract

Yannakakis' theorem relating the extension complexity of a polytope to the size of a nonnegative factorization of its slack matrix is a seminal result in the study of lifts of convex sets. Inspired by this result and the importance of lifts in the setting of integer programming, we show that a similar result holds for the discrete analog of convex polyhedral cones-affine semigroups. We define the notions of the integer slack matrix and a lift of an affine semigroup. We show that many of the characterizations of the slack matrix in the convex cone setting have analogous results in the affine semigroup setting. We also show how slack matrices of affine semigroups can be used to obtain new results in the study of nonnegative integer rank of nonnegative integer matrices.