Large-deviation Properties of Linear-programming Computational Hardness of the Vertex Cover Problem

TL;DR

This paper investigates the large-deviation properties of the computational hardness of LP relaxation for vertex cover on random graphs, revealing phase-dependent differences and their relation to graph structure and spin-glass theory.

Contribution

It introduces a rare-event sampling method to analyze the distribution of LP relaxation costs and links the hardness to phase transitions in replica symmetry.

Findings

Hardness distribution differs significantly between easy and hard problems.

Distributions are nearly indistinguishable in the RS phase but differ in the RSB phase.

A quantitative relation between graph structure and computational hardness is established.

Abstract

The distribution of the computational cost of linear-programming (LP) relaxation for vertex cover problems on Erdos-Renyi random graphs is evaluated by using the rare-event sampling method. As a large-deviation property, differences of the distribution for "easy" and "hard" problems are found reflecting the hardness of approximation by LP relaxation. In particular, by evaluating the total variation distance between conditional distributions with respect to the hardness, it is suggested that those distributions are almost indistinguishable in the replica symmetric (RS) phase while they asymptotically differ in the replica symmetry breaking (RSB) phase. In addition, we seek for a relation to graph structure by investigating a similarity to bipartite graphs, which exhibits a quantitative difference between the RS and RSB phase. These results indicate the nontrivial relation of the typical computational cost of LP relaxation to the RS-RSB phase transition as present in the spin-glass theory of models on the corresponding random graph structure.