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 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 t...

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Bibliographic Details
Published in:Theory of computing systems Vol. 61; no. 3; pp. 721 - 738
Main Authors: Shachnai, Hadas, Zehavi, Meirav
Format: Journal Article
Language:English
Published: New York Springer US 01.10.2017
Springer Nature B.V
Subjects:
Algorithms
Computer Science
Parameterization
Partitioning
Theory of Computation
Traveling salesman problem
Graph partitioning
Parameterized algorithm
Representative family
Random separation
ISSN:1432-4350, 1433-0490
Online Access:Get full text
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Summary: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 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.
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ISSN:1432-4350
1433-0490
DOI:10.1007/s00224-016-9706-0