We present experimental work on a primal-dual framework simultaneously approximating maximum cut and weighted fractional cut-covering instances. In this primal-dual framework, we solve a semidefinite programming (SDP) relaxation to either the maximum cut problem or to the weighted fractional cut-covering problem, and then independently sample a collection of cuts via the random-hyperplane technique. We then simultaneously certify the approximate optimality of a cut and a fractional cut cover. We present several implementations which reliably achieve the celebrated Goemans and Williamson approximation ratio of $α_{\mathrm{GW}} \approx 0.878$ for both optimization problems simultaneously, after $\lceil 128 \ln m \rceil$ samples, a number significantly smaller than the best theoretical bounds. This is the first experimental work approximating the weighted fractional cut-covering problem, and we deliver robust and repeatable results despite the use of randomized algorithms and floating-point arithmetic. Careful pre-processing of instances and post-processing of numeric results allow for good empirical outcomes with both first-order and second-order SDP solvers. Nearly optimal SDP solutions are suitably perturbed to ensure better probabilistic and numerical behavior. Our experiments deviate from theory by using a linear programming (LP) solver to compute fractional cut covers. For most instances studied, LP solving produces certifiably better results than the theoretical algorithm after $\lceil 128 \ln m \rceil$ samples. All our experiments strictly follow a unified pipeline which explicitly documents all parameters used in each run.

翻译：本文针对同时近似最大割和加权分数割覆盖实例的原-对偶框架开展实验研究。在该原-对偶框架中，我们求解最大割问题或加权分数割覆盖问题的半定规划（SDP）松弛，然后通过随机超平面技术独立采样一组割。随后，我们同时验证割和分数割覆盖的近似最优性。我们提出了多种实现方案，在采样次数为$\lceil 128 \ln m \rceil$（该数值显著小于最佳理论界）时，能够可靠地同时达到这两个优化问题的著名Goemans-Williamson近似比$α_{\mathrm{GW}} \approx 0.878$。这是首个针对加权分数割覆盖问题进行近似计算的实验工作，尽管采用了随机算法和浮点运算，我们仍获得了稳健且可重复的结果。通过对实例进行精心预处理并对数值结果进行后处理，使用一阶和二阶SDP求解器均能获得良好的实验效果。对接近最优的SDP解进行适当扰动，以确保更好的概率性和数值行为。我们的实验与理论有所不同：采用线性规划（LP）求解器来计算分数割覆盖。对于所研究的大多数实例，在采样次数为$\lceil 128 \ln m \rceil$后，LP求解得到的结果可验证地优于理论算法。所有实验均严格遵循统一流水线，并明确记录了每次运行中使用的所有参数。