Phase transition for cutting-plane approach to vertex-cover problem

TL;DR

This paper investigates the phase transition in the vertex-cover problem on random graphs and demonstrates that a cutting-plane algorithm exhibits an 'easy-hard' transition near the same critical point, indicating fundamental complexity features.

Contribution

It introduces a cutting-plane based algorithm for vertex cover and shows its phase transition aligns with known complexity transitions, highlighting problem-intrinsic hardness.

Findings

The algorithm exhibits an 'easy-hard' transition around c=e.

Phase transition coincides with changes in solution space structure.

Hardness is intrinsic, not dependent on problem representation.

Abstract

We study the vertex-cover problem which is an NP-hard optimization problem and a prototypical model exhibiting phase transitions on random graphs, e.g., Erdoes-Renyi (ER) random graphs. These phase transitions coincide with changes of the solution space structure, e.g, for the ER ensemble at connectivity c=e=2.7183 from replica symmetric to replica-symmetry broken. For the vertex-cover problem, also the typical complexity of exact branch-and-bound algorithms, which proceed by exploring the landscape of feasible configurations, change close to this phase transition from "easy" to "hard". In this work, we consider an algorithm which has a completely different strategy: The problem is mapped onto a linear programming problem augmented by a cutting-plane approach, hence the algorithm operates in a space OUTSIDE the space of feasible configurations until the final step, where a solution is found. Here we show that this type of algorithm also exhibits an "easy-hard" transition around c=e, which strongly indicates that the typical hardness of a problem is fundamental to the problem and not due to a specific representation of the problem.