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arXiv · 1707.04411

Isoperimetry in integer lattices

Ben Barber·Joshua Erde
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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.

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Ben Barber, Joshua Erde. 2018-04-19. Isoperimetry in integer lattices. https://doi.org/10.19086/da.3555

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