A independent set in a graph is a set of vertices that have no edges between them. How large can a independent set be in a random d-regular graph? How large can it be if we are to construct it using a (possibly randomized) algorithm that is local in nature? We will discuss a notion of local algorithms for combinatorial optimization problems on large, random d-regular graphs. We will then explain why, for asymptotically large d, local algorithms can only…

We present an extension of the Frank-Wolfe method that is designed to induce near-optimal solutions on low-dimensional faces of the feasible region. We present computational guarantees for the method that trade off efficiency in computing near-optimal solutions with upper bounds on the dimension of minimal faces of iterates. We apply our method to the low-rank matrix completion problem, where low-dimensional faces correspond to low-rank solutions. We present computational results for large-scale low-rank matrix completion problems that demonstrate significant speed-ups in…