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Approximation algorithms in combinatorial scientific computing. (English) Zbl 1440.68337
This paper conducts a thorough survey on approximation algorithms for two classical optimization problems on graphs: matching and edge cover. The problems involve the computation of degree-constrained subgraphs of a graph that might represent its significant subgraphs in order to reduce the computational costs and memory required of algorithms that obtain some useful information from the graph.
A matching is a subset of vertex-disjoint edges and we either seek to maximize the cardinality of a matching or, when weights are assigned to edges, maximize the sum of the weights of edges in a matching. The authors also discuss a less-studied variant where the weights are on the vertices instead of the edges. A generalization of matching is the \(b\)-matching problem, where we are given natural numbers \(b(v)\) for each vertex \(v\) in the graph and are required to choose at most \(b(v)\) matched edges incident on \(v\).
The edge cover problem requires to choose at least one edge incident on each vertex to belong to the edge cover. Here we seek to minimize the cardinality of the edges in the cover or the sum of weights of the edges in the cover. The generalization of edge cover leads to the \(b\)-edge cover problem where, given natural numbers \(b(v)\) for each vertex \(v\), we are required to choose at least \(b(v)\) edges incident on \(v\) to belong to the edge cover.
Exact algorithms for both problems and their variants have polynomial time complexity with the asymptotic run-time larger than the product of the number of edges times the square root of the number of vertices, and they also have a little concurrency. In order to make them practical for graphs with billions of vertices and edges, the paper also focuses on approximation algorithms with either near-linear time complexity in the size of the graph or algorithms that possess high concurrency so that they can be implemented efficiently on parallel computers.
The paper also includes a comprehensive comparison of various implementations of approximation algorithms.
MSC:
68W25 Approximation algorithms
05C07 Vertex degrees
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C85 Graph algorithms (graph-theoretic aspects)
68R10 Graph theory (including graph drawing) in computer science
68W10 Parallel algorithms in computer science
68W40 Analysis of algorithms
90C27 Combinatorial optimization
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