How to sample connected $K$-partitions of a graph

TL;DR

This paper introduces an algorithm for sampling connected K-partitions of a graph with a known probability distribution, enabling controlled partitioning of graph nodes into connected subgraphs.

Contribution

It presents a novel algorithm for sampling connected K-partitions with a closed-form probability distribution, advancing graph partitioning methods.

Findings

Algorithm efficiently samples connected K-partitions.

Closed-form expression for sampling probability derived.

Applicable to various graph partitioning problems.

Abstract

A connected undirected graph $G=(V,E)$ is given. This paper presents an algorithm that samples (non-uniformly) a $K$ partition $U_1,\ldots U_K$ of the graph nodes $V$, such that the subgraph induced by each $U_k$, with $k=1:K$, is connected. Moreover, the probability induced by the algorithm over the set ${\mathcal C}_K$ of all such partitions is obtained in closed form.