Approximate ground states of the random-field Potts model from graph cuts

TL;DR

This paper introduces a heuristic graph-cut based method to approximate ground states of the NP-hard random-field Potts model efficiently, outperforming traditional algorithms for small to moderate numbers of states.

Contribution

It presents a novel embedding technique combined with graph cuts to find approximate solutions for the Potts model's ground state efficiently.

Findings

The heuristic finds solutions comparable to quasi-exact methods in less time.

The method effectively analyzes the breakup length in 2D Potts models.

Performance is good for small to moderate number of Potts states.

Abstract

While the ground-state problem for the random-field Ising model is polynomial, and can be solved using a number of well-known algorithms for maximum flow or graph cut, the analogue random-field Potts model corresponds to a multi-terminal flow problem that is known to be NP hard. Hence an efficient exact algorithm is very unlikely to exist. As we show here, it is nevertheless possible to use an embedding of binary degrees of freedom into the Potts spins in combination with graph-cut methods to solve the corresponding ground-state problem approximately in polynomial time. We benchmark this heuristic algorithm using a set of quasi-exact ground states found for small systems from long parallel tempering runs. For not too large number $q$ of Potts states, the method based on graph cuts finds the same solutions in a fraction of the time. We employ the new technique to analyze the breakup length of the random-field Potts model in two dimensions.