Isoperimetry in integer lattices

TL;DR

This paper solves the asymptotic edge isoperimetric problem for Cayley graphs on integer lattices, identifying near-optimal shapes as zonotopes generated by the graph's generators.

Contribution

It extends the understanding of the isoperimetric problem to all Cayley graphs on integer lattices, providing asymptotic solutions and geometric characterizations.

Findings

Asymptotic solutions for all Cayley graphs on $\\mathbb Z^d$

Near-optimal shapes are zonotopes generated by Cayley graph generators

Provides geometric insights into the isoperimetric problem in lattices

Abstract

The edge isoperimetric problem for a graph $G$ is to determine, for each $n$, the minimum number of edges leaving any set of $n$ vertices. In general this problem is NP-hard, but exact solutions are known in some special cases, for example when $G$ is the usual integer lattice. We solve the edge isoperimetric problem asymptotically for every Cayley graph on $\mathbb Z^d$. The near-optimal shapes that we exhibit are zonotopes generated by line segments corresponding to the generators of the Cayley graph.