Complexity and Geometry of Sampling Connected Graph Partitions

TL;DR

This paper investigates the computational difficulty of sampling connected graph partitions, demonstrating intractability and slow mixing of practical algorithms, with implications for applications like political redistricting.

Contribution

It proves intractability results for sampling connected partitions in planar graphs and shows that the flip walk Markov chain mixes exponentially slowly in certain graph families.

Findings

Sampling is computationally intractable for certain graph classes.

The flip walk Markov chain exhibits exponential slow mixing.

Empirical evidence confirms slow mixing on grid and real-world graphs.

Abstract

In this paper, we prove intractability results about sampling from the set of partitions of a planar graph into connected components. Our proofs are motivated by a technique introduced by Jerrum, Valiant, and Vazirani. Moreover, we use gadgets inspired by their technique to provide families of graphs where the "flip walk" Markov chain used in practice for this sampling task exhibits exponentially slow mixing. Supporting our theoretical results we present some empirical evidence demonstrating the slow mixing of the flip walk on grid graphs and on real data. Inspired by connections to the statistical physics of self-avoiding walks, we investigate the sensitivity of certain popular sampling algorithms to the graph topology. Finally, we discuss a few cases where the sampling problem is tractable. Applications to political redistricting have recently brought increased attention to this problem, and we articulate open questions about this application that are highlighted by our results.