Approximating the maximally balanced connected partition problem in graphs

Chlebíková, Janka

doi:10.1016/s0020-0190(96)00175-5

Cited by 75 publications

(54 citation statements)

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“…Now we focus on the problem of Balanced 2-community in general graphs. In [8] it has been proved that to find a connected balanced partition without any additional constraints is an NP-complete problem in general graphs. We prove similar results for Balanced Weak 2-community and Balanced 2-community and their connected variants.…”

Section: Theorem 5 Let G = (V E) Be a Graph With Minimum Degreementioning

confidence: 99%

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Structural and Algorithmic Properties of 2-Community Structures

2017

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We investigate the structural and algorithmic properties of 2-community structures in graphs introduced recently by Olsen (Math Soc Sci 66(3):331-336, 2013). A 2-community structure is a partition of a vertex set into two parts such that for each vertex the numbers of neighbours in/outside its own part and the sizes of the parts are correlated. We show that some well studied graph classes as graphs of maximum degree 3, minimum degree at least |V | − 3, trees and also others, have always a 2-community structure. Furthermore, a 2-community structure can be found in polynomial time in all these classes, even with additional request of connectivity in both parts. We introduce a concept of a weak 2-community and prove that in general graphs it is NP-complete to find a balanced weak 2-community structure with or without request for connectivity in both parts. On the other hand, we present a polynomial-time algorithm to solve the problem (without the condition for connectivity of parts) in graphs of degree at most 3.

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Section: Theorem 5 Let G = (V E) Be a Graph With Minimum Degreementioning

confidence: 99%

“…due to its applications in the divide-and-conquer algorithms, see e.g. [8]. In the balanced partition problem, which can be seen as a generalisation of the bisection problem to any given number of parts, the goal is to minimise the number of edges between partitions.…”

Section: Introductionmentioning

confidence: 99%

Structural and Algorithmic Properties of 2-Community Structures

2017

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“…In (Chlebikova, 1996), authors studied the approximation of the Maximally Balanced Connected Partition problem (MBCP). They first presented the optimization problem that finds the maximally balanced connected partition for a graph G. It results in a partition (V 1 , V 2 ) of V composed of disjoint sets V 1 and V 2 such that both sub-graphs of G induced by V 1 and V 2 are connected, and maximize an objective function "balance", B w (V 1 , V 2 ) = min (w(V 1 ), w(V 2 )).…”

Section: Existing Graph Partitioning Techniquesmentioning

confidence: 99%

“…Our choice is mainly motivated by the practical approach provided in (Chlebikova, 1996) and based on the use of a polynomial-time algorithm that gives an approximate solution.…”

Section: Existing Graph Partitioning Techniquesmentioning

confidence: 99%

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Topology Control in Large Scale WSNs : Routing and Base Station Placement

Slama¹

2010

Sustainable Wireless Sensor Networks

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A new min‐cut problem with application to electric power network partitioning

Sen

Ghosh

Vittal

et al. 2008

Int Trans Elec Energy Syst

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SUMMARYThe problem of partitioning a graph into two or more subgraphs that satisfies certain conditions is encountered in many different domains. Accordingly, graph partitioning problem has been studied extensively in the last 50 years. The most celebrated result among this class of problems is the max flow ¼ min cut theorem due to Ford and Fulkerson. Utilizing the modifications suggested by Edmonds and Karp, it is well known that the minimum capacity cut in the directed graph with edge weights can be computed in polynomial time. If the partition divides the node set V into subsets V 1 and V 2 , where V 1 contains one of the specified nodes s and V 2 contains the other specified node t, the capacity of a cut is defined as the sum of the edge weights going from V 1 to V 2 . In electrical power distribution networks, a slow-coherency-based islanding strategy is used as a prevention against the cascading failures. In this paper, we concentrate on the graph partition problems which are encountered in electric power distribution networks. In this environment, two different definitions of capacity of a cut are used. In the first definition, capacity of a cut is taken to be the difference of the edge weights going from V 1 to V 2 and from V 2 to V 1 . In the second definition, the capacity of a cut is taken to be the maximum of sum of the edge weights going from V 1 to V 2 and from V 2 to V 1 . Surprisingly, with slight change of the definition of the capacity of a cut, the computational complexity of the problem changes significantly. In this paper, we show that with the new definitions of the capacity of a cut, the minimum cut computation problem becomes NP-complete. We provide an optimal solution to the problems using mathematical programming techniques. In addition, we also provide heuristic solutions and compare the performance with that of the optimal solution.

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Approximating the maximally balanced connected partition problem in graphs

Cited by 75 publications

References 1 publication

Structural and Algorithmic Properties of 2-Community Structures

Structural and Algorithmic Properties of 2-Community Structures

Topology Control in Large Scale WSNs : Routing and Base Station Placement

A new min‐cut problem with application to electric power network partitioning

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