🤖 AI Summary
This work presents the first experimental study that simultaneously approximates solutions to both the Max-Cut problem and the Weighted Fractional Cut Cover problem. Building upon a primal-dual framework, the approach integrates semidefinite programming (SDP) relaxation with randomized hyperplane rounding, while replacing conventional theoretical algorithms with a linear programming (LP) solver to substantially enhance empirical performance. Using only ⌈128 ln m⌉ random samples, the method consistently achieves an approximation ratio close to the Goemans–Williamson bound of 0.878 for the vast majority of instances, demonstrating robustness and reproducibility. Notably, the LP-based variant frequently outperforms theoretical expectations, thereby validating the efficacy and practicality of the proposed hybrid strategy.
📝 Abstract
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.