Parameterized Algorithms for Graph Partitioning Problems

Parameterized Algorithms for Graph Partitioning Problems We study a broad class of graph partitioning problems. Each problem is defined by two constants, α 1 and α 2. The input is a graph G, an integer k and a number p, and the objective is to find a subset U ⊆ V $U\subseteq V$ of size k, such that α 1 m 1 + α 2 m 2 is at most (or at least) p, where m 1, m 2 are the cardinalities of the edge sets having both endpoints, and exactly one endpoint, in U, respectively. This class of fixed-cardinality graph partitioning problems (FGPPs) encompasses Max (k, n − k)-Cut, Min k-Vertex Cover, k-Densest Subgraph, and k-Sparsest Subgraph. Our main result is a 4 k + o(k)Δ k ⋅n O(1) time algorithm for any problem in this class, where Δ ≥ 1 is the maximum degree in the input graph. This resolves an open question posed by Bonnet et al. (Proc. International Symposium on Parameterized and Exact Computation, 2013). We obtain faster algorithms for certain subclasses of FGPPs, parameterized by p, or by (k + p). In particular, we give a 4 p + o(p)⋅n O(1) time algorithm for Max (k, n − k)-Cut, thus improving significantly the best known p p ⋅n O(1) time algorithm by Bonnet et al. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Theory of Computing Systems Springer Journals

Parameterized Algorithms for Graph Partitioning Problems

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Publisher
Springer US
Copyright
Copyright © 2016 by Springer Science+Business Media New York
Subject
Computer Science; Theory of Computation
ISSN
1432-4350
eISSN
1433-0490
D.O.I.
10.1007/s00224-016-9706-0
Publisher site
See Article on Publisher Site

Abstract

We study a broad class of graph partitioning problems. Each problem is defined by two constants, α 1 and α 2. The input is a graph G, an integer k and a number p, and the objective is to find a subset U ⊆ V $U\subseteq V$ of size k, such that α 1 m 1 + α 2 m 2 is at most (or at least) p, where m 1, m 2 are the cardinalities of the edge sets having both endpoints, and exactly one endpoint, in U, respectively. This class of fixed-cardinality graph partitioning problems (FGPPs) encompasses Max (k, n − k)-Cut, Min k-Vertex Cover, k-Densest Subgraph, and k-Sparsest Subgraph. Our main result is a 4 k + o(k)Δ k ⋅n O(1) time algorithm for any problem in this class, where Δ ≥ 1 is the maximum degree in the input graph. This resolves an open question posed by Bonnet et al. (Proc. International Symposium on Parameterized and Exact Computation, 2013). We obtain faster algorithms for certain subclasses of FGPPs, parameterized by p, or by (k + p). In particular, we give a 4 p + o(p)⋅n O(1) time algorithm for Max (k, n − k)-Cut, thus improving significantly the best known p p ⋅n O(1) time algorithm by Bonnet et al.

Journal

Theory of Computing SystemsSpringer Journals

Published: Sep 5, 2016

References

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