The problem instance of Vertex Cover consists of an undirected graph G = (V, E) and a positive integer k, the question is whether there exists a subset C ⊆ V of vertices such that each edge in E has at least one of its endpoints in C with |C| ≤ k. We improve two recent worst case upper bounds for Vertex Cover. First, Balasubramanian et al. showed that Vertex Cover can be solved in time O(kn + 1...

Static Worst-Case Execution Time (WCET) analysis is a technique to derive upper bounds for the execution times of programs. Such bounds are crucial when designing and verifying real-time systems. A key component for static derivation of precise WCET estimates is upper bounds on the number of times different loops can be iterated. In this paper we present an approach for deriving upper loop boun...

Recently, a new lattice basis reduction notion, called diagonal reduction, was proposed for lattice-reduction-aided detection (LRAD) of multiinput multioutput (MIMO) systems. In this paper, we improve the efficiency of the diagonal reduction algorithm by employing the fast Givens transformations. The technique of the fast Givens is applicable to a family of LLL-type lattice reduction methods to...

We prove that Θ̃(kd2/ε2) samples are necessary and sufficient for learning a mixture of k Gaussians in Rd, up to error ε in total variation distance. This improves both the known upper bound and lower bound for this problem. For mixtures of axis-aligned Gaussians, we show that Õ(kd/ε2) samples suffice, matching a known lower bound. Moreover, these results hold in an agnostic learning setting as ...