Isoperimetry in integer 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.

Published in Discrete Analysis

References in corpus (1)

• Vertex Isoperimetric Inequalities for a Family of Graphs on Z^k

Cited by in corpus (1)

• Edge isoperimetry of lattices