COMPLEXITY AND GEOMETRY OF SAMPLING CONNECTED GRAPH PARTITIONS

Lorenzo Najt, Daryl DeFord, Justin Solomon

Abstract: In this paper, we prove intractability results about sampling from the set of partitions of a planar graph intoconnected components. Our proofs are motivated by a technique introduced by Jerrum, Valiant, and Vazirani. Moreover, weuse gadgets inspired by their technique to provide families of graphs where the “flip walk” Markov chain used in practice forthis sampling task exhibits exponentially slow mixing. Supporting our theoretical results we present some empirical evidencedemonstrating the slow mixing of the flip walk on grid graphs and on real data. Inspired by connections to the statisticalphysics 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 recentlybrought increased attention to this problem, and we articulate open questions about this application that are highlighted byour results.