Marcel K. de Carli Silva

Marcel K. de Carli Silva, Nicholas J. A. Harvey, Cristiane M. Sato

Recently there has been much interest in "sparsifying" sums of rank one matrices: modifying the coefficients such that only a few are nonzero, while approximately preserving the matrix that results from the sum. Results of this sort have found applications in many different areas, including sparsifying graphs. In this paper we consider the more general problem of sparsifying sums of positive semidefinite matrices that have arbitrary rank. We give several algorithms for solving this problem. The first algorithm is based on the method of Batson, Spielman and Srivastava (2009). The second algorithm is based on the matrix multiplicative weights update method of Arora and Kale (2007). We also highlight an interesting connection between these two algorithms. Our algorithms have numerous applications. We show how they can be used to construct graph sparsifiers with auxiliary constraints, sparsifiers of hypergraphs, and sparse solutions to semidefinite programs.

Nathan Benedetto Proença, Marcel K. de Carli Silva, Cristiane M. Sato et al.

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.