On the replica symmetry phase of the independent set problem

TL;DR

This paper rigorously analyzes the replica symmetry phase of the independent set problem on Erdős-Rényi graphs using statistical mechanics, establishing the free energy and conjecturing the ground state structure.

Contribution

It introduces a Hamiltonian framework for the ISP, rigorously proves the existence of a replica symmetry phase, and derives a conjectural formula for the ground state based on Parisi theory.

Findings

Established the free energy in the replica symmetry phase

Proved the existence of a replica symmetry phase rigorously

Derived a conjectural formula for the ground state

Abstract

The independent set problem, ISP for short, asks for the maximal number of vertices in a (large) graph which can be occupied such that none of them are neighbors. We address the question from a statistical mechanics perspective, in the case of Erdoes-Renyi random graphs. We thereby introduce a Hamiltonian penalizing configurations which do not satisfy the non-neighboring constraint: the ground state of the ensuing disordered system corresponds to the solution of the ISP. Identifying the ground state amounts, in turns, to control the phase where replica symmetry is broken, which is way beyond our current understanding. By means of Talagrand's cavity method, we rigorously establish the existence of a replica symmetry phase, computing, in particular, the free energy in the limit of large graphs. A conjectural formula for the ground state, hence for the solution of the ISP, is also derived. Being based on the Parisi theory, the emerging picture is that of a staggering complexity.