We survey recent work on approximation algorithms for computing degreeconstrained subgraphs in graphs and their applications in combinatorial scientific computing. The problems we consider include maximization versions of cardinality matching, edgeweighted matching, vertexweighted matching and edgeweighted $b$ matching, and minimization versions of weighted edge cover and $b$ edge cover. Exact algorithms for these problems are impractical for massive graphs with several millions of edges. For each problem we discuss theoretical foundations, the design of several linear or nearlinear time approximation algorithms, their implementations on serial and parallel computers, and applications. Our focus is on practical algorithms that yield good performance on modern computer architectures with multiple threads and interconnected processors. We also include information about the software available for these problems.
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A Parallel Approximation Algorithm for Maximizing Submodular bMatching
We design new serial and parallel approximation algorithms for computing a maximum weight bmatching in an edgeweighted graph with a submodular objective function. This problem is NPhard; the new algorithms have approximation ratio 1/3, and are relaxations of the Greedy algorithm that rely only on local information in the graph, making them parallelizable. We have designed and implemented Local Lazy Greedy algorithms for both serial and parallel computers. We have applied the approximate submodular bmatching algorithm to assign tasks to processors in the computation of Fock matrices in quantum chemistry on parallel computers. The assignment seeks to reduce the run time by balancing the computational load on the processors and bounding the number of messages that each processor sends. We show that the new assignment of tasks to processors provides a four fold speedup over the currently used assignment in the NWChemEx software on 8000 processors on the Summit supercomputer at Oak Ridge National Lab.
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 Award ID(s):
 1637534
 NSFPAR ID:
 10300888
 Editor(s):
 Bender, M.; Gilbert, J.; Hendrickson, B.; Sullivan, B.
 Date Published:
 Journal Name:
 Proceedings of the 2021 SIAM Conference on Applied and Computational Discrete Algorithms (ACDA21)
 Volume:
 1
 Issue:
 1
 Page Range / eLocation ID:
 4556
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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