We present a simple distributed @math -approximation algorithm for maximum weight independent set (MaxIS) in the @math model which completes in @math rounds, where @math is the maximum degree, @math is the number of rounds needed to compute a maximal independent set (MIS) on @math , and @math is the maximum weight of a node. Whether our algorithm is randomized or deterministic depends on the MIS algorithm used as a black-box. Plugging in the best known algorithm for MIS gives a randomized solution in @math rounds, where @math is the number of nodes. We also present a deterministic @math -round algorithm based on coloring. We then show how to use our MaxIS approximation algorithms to compute a @math -approximation for maximum weight matching without incurring any additional round penalty in the @math model. We use a known reduction for simulating algorithms on the line graph while incurring congestion, but we show our algorithm is part of a broad family of for which we describe a mechanism that allows the simulation to run in the @math model without an additional overhead. Next, we show that for maximum weight matching, relaxing the approximation factor to ( @math ) allows us to devise a distributed algorithm requiring @math rounds for any constant @math . For the unweighted case, we can even obtain a @math -approximation in this number of rounds. These algorithms are the first to achieve the provably optimal round complexity with respect to dependency on @math .
The first distributed algorithm that uses the local ratio technique is due to @cite_7 . The local ratio technique was also used in @cite_34 to compute a distributed @math -approximation for weighted vertex cover. In @cite_12 , a similar technique of weight grouping is used in the primal-dual framework for scheduling.
@cite_34 @cite_12 @cite_7
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We obtain improved algorithms for finding small vertex covers in bounded degree graphs and hypergraphs. We use semidefinite programming to relax the problems and introduce new rounding techniques for these relaxations. On graphs with maximum degree at most @math , the algorithm achieves a performance ratio of @math for large @math , which improves the previously known ratio of @math obtained by Halld orsson and Radhakrishnan. Using similar techniques, we also present improved approximations for the vertex cover problem in hypergraphs. For k-uniform hypergraphs with n vertices, we achieve a ratio of @math for large n, and for k-uniform hypergraphs with maximum degree at most @math the algorithm achieves a ratio of @math for large @math . These results considerably improve the previous best ratio of @math for bounded degree k-uniform hypergraphs, and @math for general k-uniform hypergraphs, both obtained by Krivelevich. Using similar techniques, we also obtain an approximation algorithm for the weighted independent set problem, matching a recent result of Halldorsson. In this paper we give an efficient distributed algorithm computing approximate solutions to a very general, and classical, scheduling problem. The approximation guarantee is within a constant factor of the optimum. By "efficient", we mean that the number of communication rounds is poly-logarithmic in the size of the input. In the problem, we have a bipartite graph with computing agents on one side and resources on the other. Agents that share a resource can communicate in one time step. Each agent has a list of jobs, each with its own length and profit, to be executed on a neighbouring resource within a given time-window. Resources can execute non preemptively only one job at a time. The goal is to maximize the profit of the jobs that are scheduled. It is well known that this problem is NP-hard. A very interesting feature of our algorithm is that it is derived in a systematic manner from a primal-dual algorithm.