A spin glass approach to the directed feedback vertex set problem

TL;DR

This paper models the directed feedback vertex set problem using a spin glass approach, applying statistical physics methods and a belief propagation algorithm to improve solution quality on large graphs.

Contribution

It introduces a novel spin glass model for the directed FVS problem and develops a belief propagation-guided decimation algorithm for better solutions.

Findings

BPD slightly outperforms simulated annealing on large graphs

RS mean field theory predictions are lower than BPD results

Cycle-caused correlations are neglected in the RS approximation

Abstract

A directed graph (digraph) is formed by vertices and arcs (directed edges) from one vertex to another. A feedback vertex set (FVS) is a set of vertices that contains at least one vertex of every directed cycle in this digraph. The directed feedback vertex set problem aims at constructing a FVS of minimum cardinality. This is a fundamental cycle-constrained hard combinatorial optimization problem with wide practical applications. In this paper we construct a spin glass model for the directed FVS problem by converting the global cycle constraints into local arc constraints, and study this model through the replica-symmetric (RS) mean field theory of statistical physics. We then implement a belief propagation-guided decimation (BPD) algorithm for single digraph instances. The BPD algorithm slightly outperforms the simulated annealing algorithm on large random graph instances. The predictions of the RS mean field theory are noticeably lower than the BPD results, possibly due to its neglect of cycle-caused long range correlations.