Cutting-Plane Algorithms and Solution Whitening for the Vertex-Cover Problem

TL;DR

This paper investigates the phase transition and algorithmic boundaries of the NP-hard vertex-cover problem on random graphs using linear programming and cutting-plane methods, revealing new transitions and insights into solution structure.

Contribution

It introduces cutting-plane techniques to improve LP solutions for vertex cover, identifying a new algorithmic transition beyond the known RSB point.

Findings

Cutting-plane methods extend the boundary of complete solutions beyond the RSB transition.

Evidence of a new algorithmic transition at average degree c=2.90.

Whitening measures show slight but measurable effects of the RSB transition.

Abstract

The phase-transition behavior of the NP-hard vertex-cover (VC) combinatorial optimization problem is studied numerically by linear programming (LP) on ensembles of random graphs. As the basic Simplex (SX) algorithm suitable for such LPs may produce incomplete solutions for sufficiently complex graphs, the application of cutting-plane (CP) methods is sought. We consider Gomory and {0,1/2} cuts. We measure the probability of obtaining complete solutions with these approaches as a function of the average node degree c and observe transition between typically complete and incomplete phase regions. While not generally complete solutions are obtained for graphs of arbitrarily high complexity, the CP approaches still advance the boundary in comparison to the pure SX algorithm, beyond the known replica-symmetry breaking (RSB) transition at c=e=2.718... . In fact, our results provide evidence for another algorithmic transition at c=2.90(2). Besides this, we quantify the transition between easy and hard solvability of the VC problem also in terms of numerical effort. Further we study the so-called whitening of the solution, which is a measure for the degree of freedom that single vertices experience with respect to degenerate solutions. Inspection of the quantities related to clusters of white vertices reveals that whitening is affected, only slightly but measurably, by the RSB transition.