![Counting Complexity and Phase Transitions_hi-res logo](/sites/default/files/styles/workshop_banner_sm_1x/public/2023-01/Counting%20Complexity%20and%20Phase%20Transitions_hi-res.jpg?h=bf33d09a&itok=MrH5eN5T)
Abstract
The Max-Cut problem seeks to determine the maximal cut size in a given graph. With no polynomial-time efficient approximation for Max-Cut (unless P=NP), its asymptotic for a typical large sparse graph is of considerable interest. We prove that for uniformly random d-regular graph of N vertices, and for the uniformly chosen Erdos-Renyi graph of M=N d/2 edges, the leading correction to M/2 (the typical cut size), is P_* sqrt(N M/2). Here P_* is the ground state energy of the Sherrington-Kirkpatrick model, expressed analytically via Parisi's formula.
This talk is based on a joint work with Subhabrata Sen and Andrea Montanari.