Pull-Push Method: A new approach to Edge-Isoperimetric Problems

TL;DR

This paper introduces the pull-push method, a novel technique for solving edge-isoperimetric problems, generalizing previous results and partially addressing Harper's question on local-global principles.

Contribution

The paper presents the pull-push method, a new approach that unifies and extends existing results in edge-isoperimetric problems, and offers a strategy to include grid cases.

Findings

Generalized the Ahlswede-Cai local-global principle

Introduced the pull-push technique for edge-isoperimetric problems

Partially answered Harper's question on local-global principles

Abstract

We prove a generalization of the Ahlswede-Cai local-global principle. A new technique to handle edge-isoperimetric problems is introduced which we call the pull-push method. Our main result includes all previously published results in this area as special cases with the only exception of the edge-isoperimetric problem for grids. With this we partially answer a question of Harper on local-global principles. We also describe a strategy for further generalization of our results so that the case of grids would be covered, which would completely settle Harper's question.