Knapsack Cover Subject to a Matroid Constraint

We consider the Knapsack Covering problem subject to a matroid constraint. In this problem, we are given an universe U of n items where item i has attributes: a cost c(i) and a size s(i). We also have a demand D. We are also given a matroid M = (U, I) on the set U. A feasible solution S to the problem is one such that (i) the cumulative size of the items chosen is at least D, and (ii) the set S is independent in the matroid M (i.e. S is in I). The objective is to minimize the total cost of the items selected, sum_{i in S}c(i). Our main result proves a 2-factor approximation for this problem. The problem described above falls in the realm of mixed packing covering problems. We also consider packing extensions of certain other covering problems and prove that in such cases it is not possible to derive any constant factor pproximations

Quadratic and Near-Quadratic Lower Bounds for the CONGEST Model

We present the first super-linear lower bounds for natural graph problems in the CONGEST model, answering a long-standing open question. Specifically, we show that any exact computation of a minimum vertex cover or a maximum independent set requires a near-quadratic number of rounds in the CONGEST model, as well as any algorithm for computing the chromatic number of the graph. We further show that such strong lower bounds are not limited to NP-hard problems, by showing two simple graph problems in P which require a quadratic and near-quadratic number of rounds. Finally, we address the problem of computing an exact solution to weighted all-pairs-shortest-paths (APSP), which arguably may be considered as a candidate for having a super-linear lower bound. We show a simple linear lower bound for this problem, which implies a separation between the weighted and unweighted cases, since the latter is known to have a sub-linear complexity. We also formally prove that the standard Alice-Bob framework is incapable of providing a super-linear lower bound for exact weighted APSP, whose complexity remains an intriguing open question

A primal-dual bicriteria distributed algorithm for capacitated vertex cover

In this paper we consider the capacitated vertex cover problem, which is the variant of vertex cover where each node is allowed to cover a limited number of edges. We present an efficient, deterministic, distributed approximation algorithm for the problem. Our algorithm computes a (2+epsilon)-approximate solution which violates the capacity constraints by a factor of (4+epsilon) in a polylogarithmic number of communication rounds. On the other hand, we also show that every efficient distributed approximation algorithm for this problem must violate the capacity constraints. Our result is achieved in two steps. We first develop a 2-approximate, sequential primal-dual algorithm that violates the capacity constraints by a factor of 2. Subsequently, we present a distributed version of this algorithm. We demonstrate that the sequential algorithm has an inherent need for synchronization which forces any naive distributed implementation to use a linear number of communication rounds. The challenge in this step is therefore to achieve a reduction of the communication complexity to a polylogarithmic number of rounds without worsening the approximation guarantee