A Graph Decomposition motivated by the Geometry of Randomized Rounding

TL;DR

This paper introduces a universal graph decomposition inspired by the geometry of a dynamical system related to randomized rounding, with implications for judicious partitions and MaxCut heuristics.

Contribution

It presents a new graph decomposition framework motivated by the geometry of randomized rounding, connecting it to judicious partitions and MaxCut heuristics.

Findings

Decomposition exists for all simple, connected graphs.

Vertices are partitioned into A, B, C with specific neighbor properties.

Connections to MaxCut and judicious partitions are established.

Abstract

We introduce a graph decomposition which exists for all simple, connected graphs $G=(V,E)$. The decomposition $V = A \cup B \cup C$ is such that each vertex in $A$ has more neighbors in $B$ than in $A$ and vice versa. $C$ is `balanced': each $v \in C$ has the same number of neighbours in $A$ and $B$. These decompositions arise naturally from the behavior of an associated dynamical system (`Randomized Rounding') on $(\mathbb{S}^1)^{|V|}$. Connections to judicious partitions and the \textsc{MaxCut} problem (in particular the Burer-Monteiro-Zhang heuristic) are being discussed.