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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Approximation algorithms in combinatorial scientific computing
We survey recent work on approximation algorithms for computing degree-constrained subgraphs in graphs and their applications in combinatorial scientific computing. The problems we consider include maximization versions of cardinality matching, edge-weighted matching, vertex-weighted matching and edge-weighted $b$ -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 near-linear 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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- Award ID(s):
- 1637534
- PAR ID:
- 10109987
- Date Published:
- Journal Name:
- Acta Numerica
- Volume:
- 28
- ISSN:
- 0962-4929
- Page Range / eLocation ID:
- 541 to 633
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/S0962492919000035
