NC Approximation Algorithm Research Articles

AN EFFICIENT NC ALGORITHM FOR APPROXIMATE MAXIMUM WEIGHT MATCHING

We present an alternative implementation of the (1 – ϵ) factor NC approximation algorithm for the maximum weight matching by Hougardy et al. [1]. Our implementation, on the EREW PRAM model of computation, achieves an [Formula: see text] factor improvement on both the execution time and the number of processors.

Approximating weighted matchings in parallel

We present an NC approximation algorithm for the weighted matching problem in graphs with an approximation ratio of ( 1 − ε ) . This improves the previously best approximation ratio of ( 1 2 − ε ) of an NC algorithm for this problem.

Approximating Unweighted Connectivity Problems in Parallel

Given an integer k and a k-edge-connected graph G=(V, E), we wish to find an E′⊆E of minimum size such that the graph (V, E′) is k-edge-connected. This problem is NP-hard and the best performance ratio achieved by known NC approximation algorithms is 2. For the special case where the input integer k is fixed to be 2, it is known that a performance ratio of 1.5+ϵ for any ϵ>0 can be achieved by an NC approximation algorithm. This paper considers the more general case where k is polylogarithmic in the size of the input graph, and presents the first NC approximation algorithm with a performance ratio of 1.924 for this case. We also consider the vertex analogue of this problem in which we require k-vertex-connectivity instead of k-edge-connectivity. We present the first NC approximation algorithm with a performance ratio of 1.931 for the special case where the input integer k is fixed to be 3.

Parallel approximation algorithms for maximum weighted matching in general graphs

The problem of computing a matching of maximum weight in a given edge-weighted graph is not known to be P -hard or in RNC . This paper presents two parallel approximation algorithms for this problem. The first is an RNC -approximation scheme, i.e., an RNC algorithm that computes a matching of weight at least 1−ε times the maximum for any fixed constant ε>0. The other is an NC approximation algorithm achieving an approximation ratio of 1/(2+ε) for any fixed constant ε>0.

Fast RNC and NC algorithms for maximal path sets

We present two parallel algorithms for finding a maximal set of paths in a given undirected graph. One is randomized and runs in O( log n) expected time with O( n + m) processors on a CRCW PRAM. The other is deterministic and runs in O( log 2 n) time with O( Δ 2(n + m) log n ) processors on an EREW PRAM. The results improve on the previous bests and can also be extended to digraphs. We then use the results to improve the time complexity of the best previous NC approximation algorithm for the shortest superstring problem.

Efficient NC algorithms for set cover with applications to learning and geometry

In this paper we give the first NC approximation algorithms for the unweighted and weighted set cover problems. Our algorithms use a linear number of processors and give a cover that has at most log n times the optimal size/weight, thus matching the performance of the best sequential algorithms. We apply our set cover algorithm to learning theory, giving an NC algorithm to learn the concept class obtained by taking the closure under finite union or finite intersection of any concept class of finite VC-dimension that has an NC hypothesis finder. In addition, we give a linear-processor NC algorithm for a variant of the set cover problem first proposed by Chazelle and Friedman and use it to obtain NC algorithms for several problems in computational geometry.