Cutting-plane algorithms and solution whitening for the vertex-cover problem,Physical Review E

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Cutting-plane algorithms and solution whitening for the vertex-cover problem
Physical Review E ( IF 2.2 ) Pub Date : 2022-09-19 , DOI: 10.1103/physreve.106.035305
G Claussen ₁ , A K Hartmann ₁

Affiliation

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, \frac{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 \approx 2.718

. In fact, our results provide evidence for another algorithmic transition at

c \approx 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.

中文翻译：

顶点覆盖问题的切割平面算法和解决方案白化

NP-hard vertex-cover (VC) 组合优化问题的相变行为通过对随机图集合的线性规划 (LP) 进行数值研究。由于适用于此类 LP 的基本 Simplex (SX) 算法可能会为足够复杂的图产生不完整的解决方案，因此寻求切割平面 (CP) 方法的应用。我们考虑 Gomory 和