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Digital Hybrid Model Preparation for Virtual Planning of Reconstructive Dentoalveolar Surgical Procedures

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

Timo Dewenter¹, Alexander K Hartmann

¹Institut für Physik, Universität Oldenburg, D-26111 Oldenburg, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|December 11, 2012

Summary

The vertex-cover problem

Area of Science:

Computational complexity theory
Statistical physics of complex systems
Discrete mathematics

Background:

The vertex-cover problem is a classic NP-hard optimization problem.
Phase transitions in random graphs, like Erdős-Rényi (ER) graphs, are linked to solution space structure changes.
Algorithm complexity for vertex cover shifts from easy to hard near the c=e phase transition.

Purpose of the Study:

To investigate the computational hardness of the vertex-cover problem using a novel algorithmic approach.
To determine if the observed easy-hard transition is inherent to the problem or specific to certain algorithms.

Main Methods:

Mapping the vertex-cover problem to a linear programming problem.
Employing a cutting-plane approach within the linear programming framework.

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Analyzing the algorithm's performance and complexity around the phase transition point (c=e).

Main Results:

The cutting-plane algorithm, operating outside the feasible configuration space, also exhibits an easy-hard transition near c=e.
This transition occurs despite the algorithm's fundamentally different strategy compared to traditional methods.

Conclusions:

The easy-hard transition in the vertex-cover problem's computational complexity appears to be fundamental.
Problem hardness is likely intrinsic and not solely dependent on the specific algorithmic representation or solution space exploration method.