Koopman analysis of combinatorial optimization problems with replica exchange Monte Carlo method

TL;DR

This paper applies Koopman analysis to data from replica exchange Monte Carlo methods to identify indicators of problem difficulty in combinatorial optimization, providing insights into solution search effectiveness.

Contribution

It introduces a new aggregated quantity derived from Koopman analysis of multiple time-series data to assess problem difficulty.

Findings

Negative correlation between the proposed quantity and solution search ability

Numerical experiments validate the effectiveness of the indicator

Provides a physics-inspired approach to evaluate combinatorial problem hardness

Abstract

Combinatorial optimization problems play crucial roles in real-world applications, and many studies from a physics perspective have contributed to specialized hardware for high-speed computation. However, some combinatorial optimization problems are easy to solve, and others are not. Hence, the qualification of the difficulty in problem-solving will be beneficial. In this paper, we employ the Koopman analysis for multiple time-series data from the replica exchange Monte Carlo method. After proposing a quantity that aggregates the information of the multiple time-series data, we performed numerical experiments. The results indicate a negative correlation between the proposed quantity and the ability of the solution search.