Spin glass approach to the feedback vertex set problem

TL;DR

This paper introduces a spin glass model for the feedback vertex set problem and employs belief propagation algorithms to find near-optimal solutions on random graphs and lattices, with applications in understanding complex networks.

Contribution

It develops a novel spin glass formulation for the FVS problem and demonstrates effective message-passing algorithms for large, complex networks.

Findings

Nearly optimal FVS solutions on random graphs

Effective belief propagation-guided decimation algorithm

Model applicable to directed graphs and network complexity analysis

Abstract

A feedback vertex set (FVS) of an undirected graph is a set of vertices that contains at least one vertex of each cycle of the graph. The feedback vertex set problem consists of constructing a FVS of size less than a certain given value. This combinatorial optimization problem has many practical applications, but it is in the nondeterministic polynomial-complete class of worst-case computational complexity. In this paper we define a spin glass model for the FVS problem and then study this model on the ensemble of finite-connectivity random graphs. In our model the global cycle constraints are represented through the local constraints on all the edges of the graph, and they are then treated by distributed message-passing procedures such as belief propagation. Our belief propagation-guided decimation algorithm can construct nearly optimal feedback vertex sets for single random graph instances and regular lattices. We also design a spin glass model for the FVS problem on a directed graph. Our work will be very useful for identifying the set of vertices that contribute most significantly to the dynamical complexity of a large networked system.