We describe a 3/2-approximation algorithm, \lse, for computing a b-edgecover
of minimum weight in a graph with weights on the edges.
The b-edgecover problem is a generalization of the better-known Edge Cover problem in graphs,
where the objective is to choose a subset C of edges in the graph such that
at least a specified number b(v) of edges in C are incident on each vertex v.
In the weighted b-edgecover problem, we minimize the sum of the weights of the edges in C.
We prove that the Locally Subdominant edge (LSE) algorithm computes the same b-edge cover as the one obtained by the Greedy algorithm for the problem.
However, the Greedy algorithm requires edges to be sorted
by their effective weights, and these weights need to be updated after each iteration.
These requirements make the Greedy algorithm sequential and impractical for massive graphs.
The LSE algorithm avoids the sorting step, and is amenable for parallelization.
We implement the algorithm on a serial machine and compare its performance against a
collection of approximation algorithms for the b-edge cover problem.
Our results show that the algorithm is 3 to 5 times faster than the
Greedy algorithm on a serial processor.
The approximate edge covers obtained by the LSE algorithm have weights
greater by at most 17% of the optimal weight for problems where
we could compute the latter.
We also investigate the relationship between the b-edge cover and the b-matching problems,
show that the latter has a faster implementation since edge weights are static in this
algorithm, and obtain a heuristic solution for the former from the latter.
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A Parallel Approximation Algorithm for Maximizing Submodular b-Matching
We design new serial and parallel approximation algorithms for computing a maximum weight b-matching in an edge-weighted graph with a submodular objective function. This problem is NP-hard; 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 b-matching 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
- NSF-PAR 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:
- 45-56
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1137/1.9781611976830.5
