Maximum Cuts and Fractional Cut Covers: A Computational Study of a Randomized Semidefinite Programming Approach

TL;DR

This paper experimentally evaluates a primal-dual SDP-based approach for approximating maximum cut and fractional cut-covering problems, achieving near-optimal ratios with fewer samples than theoretical bounds.

Contribution

First experimental study applying a randomized SDP approach to the weighted fractional cut-covering problem, demonstrating reliable results and practical improvements over theoretical bounds.

Findings

Achieved the Goemans-Williamson approximation ratio of approximately 0.878 for both problems.

Used fewer samples than the theoretical upper bounds, specifically $ ceil 128 imes ext{ln}(m) ceil$ samples.

LP solvers often produced better results than the theoretical algorithms in practice.

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 $\alpha_{\mathrm{GW}} \approx 0.878$ 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.