We describe a paradigm for designing parallel algorithms via approximation, and illustrate it on the bedgecover problem.
A bedgecover of minimum weight in a graph
is a subset $C$ of its edges such that
at least a specified number $b(v)$ of edges in $C$ is incident on each vertex $v$, and the sum of the edge weights in $C$ is minimum.
The Greedy algorithm and a variant, the LSE algorithm, provide $3/2$approximation guarantees in the worstcase for this problem, but these algorithms have limited parallelism. Hence we design two new $2$approximation algorithms with greater concurrency. The MCE algorithm reduces the computation of a bedgecover to that of finding a b'matching, by exploiting the relationship between these subgraphs in an approximation context.
The LSENW is derived from the LSEalgorithm using static edge weights rather than dynamically computing effective edge weights. This relaxation gives LSE a worse approximation guarantee but makes it more amenable to parallelization. We prove that both the MCE and LSENW algorithms compute the same bedgecover with at most twice the weight of the minimum weight edge cover. In practice,
the $2$approximation and $3/2$approximation algorithms compute edge covers of weight within $10\%$ the optimal. We implement three of the approximation algorithms, MCE, LSE, and LSENW on shared memory multicore machines, including an Intel Xeon and an IBM Power8 machine with 8 TB memory. The MCE algorithm is the fastest of these by an order of magnitude or more. It computes an edge cover in a graph with billions of edges in $20$ seconds using two hundred threads on the IBM Power8. We also show that the parallel depth and work can be bounded for the Suitor and bSuitor algorithms when edge weights are random.
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New Approximation Algorithms for Minimum Weighted Edge Cover
We describe two new 3/2approximation algorithms and a new 2approximation algorithm for the minimum weight edge cover problem in graphs. We show that one of the 3/2approximation algorithms, the Dual cover algorithm, computes the lowest weight edge cover relative to previously known algorithms as well as the new algorithms reported here. The Dual cover algorithm can also be implemented to be faster than the other 3/2approximation algorithms on serial computers. Many of these algorithms can be extended to solve the 6Edge cover problem as well. We show the relation of these algorithms to the KNearest Neighbor graph construction in semisupervised learning and other applications.
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 Award ID(s):
 1637534
 NSFPAR ID:
 10109986
 Date Published:
 Journal Name:
 2018 Proceedings of the Eighth SIAM Workshop on Combinatorial Scientific Computing
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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