We develop a dynamic version of the primal-dual method for optimization problems, and apply it to obtain the following results. (1) For the dynamic set-cover problem, we maintain an O(f)approximately optimal solution in O(f · log(m + n)) amortized update time, where f is the maximum “frequency” of an element, n is the number of sets, andm is the maximum number of elements in the universe at any...

The input to the minimum latency set cover problem (MLSC) consists of a set of jobs and a set of tools. Each job j needs a specific subset Sj of the tools in order to be processed. It is possible to install a single tool in every time unit. Once the entire subset Sj has been installed, job j can be processed instantly. The problem is to determine an order of job installations which minimizes th...

We develop the first streaming algorithm and the first two-party communication protocol that uses a constant number of passes/rounds and sublinear space/communication for logarithmic approximation to the classic Set Cover problem. Specifically, for n elements andm sets, our algorithm/protocol achieves a space bound of O(m · n log n logm) using O(4) passes/rounds while achieving an approximation...

1) Motivationg ◦Set Cover Given a set S with m elements and a set {Ci} of n subsets of S each one having a cost ci, the set cover problem requires to find a selection of subsets of S that cover the whole set and which have minimum cost. ◦GSEMO multiobjective EA on a 2-objective function (cost, uncovered elements). From Friedrich et al. (2010) approximation ratio O(logm) in polynomial time. How ...

Given a finite set of straight line segments S in R and some k ∈ N, is there a subset V of points on segments in S with |V | ≤ k such that each segment of S contains at least one point in V ? This is a special case of the set covering problem where the family of subsets given can be taken as a set of intersections of the straight line segments in S. Requiring that the given subsets can be inter...

We study the relationship between the approximation factor for the Set-Cover problem and the parameters ∆ : the maximum cardinality of any subset, and k : the maximum number of subsets containing any element of the ground set. We show an LP rounding based approximation of (k− 1)(1− e ln∆ k−1 ) + 1, which is substantially better than the classical algorithms in the range k ≈ ln∆, and also improv...

We address a version of the set-cover problem where we do not know the sets initially (and hence referred to as covert) but we can query an element to find out which sets contain this element as well as query a set to know the elements. We want to find a small set-cover using a minimal number of such queries. We present a Monte Carlo randomized algorithm that approximates an optimal set-cover o...

Recently, a new approach to get a deeper understanding of online computation has been introduced: the study of the advice complexity of online problems. The idea is to measure the information that online algorithms need to be supplied with to compute high-quality solutions and to overcome the drawback of not knowing future input requests. This concept was successfully applied to many prominent ...

Here we consider using quantum annealing to solve Set Cover with Pairs (SCP), an NP-hard combinatorial optimization problem that plays an important role in networking, computational biology, and biochemistry. We show an explicit construction of Ising Hamiltonians whose ground states encode the solution of SCP instances. We numerically simulate the time-dependent Schrödinger equation in order to...