Ground-State Entropy of the Random Vertex-Cover Problem

TL;DR

This paper introduces a cavity method-based iterative approach to estimate the ground-state entropy of the random vertex-cover problem, overcoming divergence issues in traditional spin-glass theories and applicable to various random-graph systems.

Contribution

The paper presents a novel cavity method algorithm to accurately estimate ground-state entropy in random vertex-cover problems, even when 1RSB theory is unstable.

Findings

Successfully estimates ground-state entropy for random graphs.

Overcomes divergence issues in zero-temperature 1RSB spin-glass theory.

Method applicable to other random-graph spin-glass systems.

Abstract

Counting the number of ground states for a spin-glass or NP-complete combinatorial optimization problem is even more difficult than the already hard task of finding a single ground state. In this paper the entropy of minimum vertex-covers of random graphs is estimated through a set of iterative equations based on the cavity method of statistical mechanics. During the iteration both the cavity entropy contributions and cavity magnetizations for each vertex are updated. This approach overcomes the difficulty of iterative divergence encountered in the zero temperature first-step replica-symmetry-breaking (1RSB) spin-glass theory. It is still applicable when the 1RSB mean-field theory is no longer stable. The method can be extended to compute the entropies of ground-states and metastable minimal-energy states for other random-graph spin-glass systems.