Entropy Inflection and Invisible Low-Energy States: Defensive Alliance Example

TL;DR

This paper reveals that traditional physics-inspired optimization methods like simulated annealing can fail due to entropy inflection points, and proposes an energy-clamping strategy to find better solutions in complex problems.

Contribution

It uncovers a bifurcation in the entropy-energy profile of the defensive alliance problem and introduces an energy-clamping method to overcome SA limitations.

Findings

Identifies a bifurcation caused by entropy inflection points.

Shows SA cannot follow the low-energy branch across the phase transition.

Proposes an energy-clamping strategy to find superior solutions.

Abstract

Lower temperature leads to a higher probability of visiting low-energy states. This intuitive belief underlies most physics-inspired strategies for addressing hard optimization problems. For instance, the popular simulated annealing (SA) dynamics is expected to approach a ground state if the temperature is lowered appropriately. Here we demonstrate that this belief is not always justified. Specifically, we employ the cavity method to analyze the minimum strong defensive alliance problem and discover a bifurcation in the solution space, induced by an inflection point in the entropy--energy profile. While easily accessible configurations are associated with the lower-free-energy branch, the low-energy configurations are associated with the higher-free-energy branch within the same temperature range. There is a discontinuous phase transition between the high-energy configurations and the ground states, which generally cannot be followed by SA. We introduce an energy-clamping strategy to obtain superior solutions by following the higher-free-energy branch, overcoming the limitations of SA.