Imbalance is fixed parameter tractable

In the Imbalance Minimization problem we are given a graph G=(V,E) and an integer b and asked whether there is an ordering v1…vn of V such that the sum of the imbalance of all the vertices is at most b. The imbalance of a vertex vi is the absolute value of the difference between the number of neighb...

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Vydáno v:	Information processing letters Ročník 113; číslo 19-21; s. 714 - 718
Hlavní autoři:	Lokshtanov, Daniel, Misra, Neeldhara, Saurabh, Saket
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Amsterdam Elsevier B.V 01.09.2013 Elsevier Sequoia S.A
Témata:	Algorithms Balancing Computer science Data processing Fixed-parameter tractable algorithm Graph layout problems Graph theory Graphs Imbalance Integer programming Lectures Mathematical problems Numerical analysis Optimization Order disorder Parameterized complexity Studies Graph layout problems Algorithms Imbalance Parameterized complexity Fixed-parameter tractable algorithm
ISSN:	0020-0190, 1872-6119
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Abstract	In the Imbalance Minimization problem we are given a graph G=(V,E) and an integer b and asked whether there is an ordering v1…vn of V such that the sum of the imbalance of all the vertices is at most b. The imbalance of a vertex vi is the absolute value of the difference between the number of neighbors to the left and right of vi. The problem is also known as the Balanced Vertex Ordering problem and it finds many applications in graph drawing. We show that this problem is fixed parameter tractable and provide an algorithm that runs in time 2O(blogb)⋅nO(1). This resolves an open problem of Kára et al. [On the complexity of the balanced vertex ordering problem, in: COCOON, in: Lecture Notes in Comput. Sci., vol. 3595, 2005, pp. 849–858]. •The Imbalance Minimization problem on graphs asks for an ordering of the vertices such that the total imbalance is minimized.•The problem finds several applications in graph drawing, and is known to be NP-hard.•In this article, we show that the problem is fixed parameter tractable, when parameterized by the solution size, resolving an open question from the literature.
AbstractList	In the Imbalance Minimization problem we are given a graph G=(V,E) and an integer b and asked whether there is an ordering v1…vn of V such that the sum of the imbalance of all the vertices is at most b. The imbalance of a vertex vi is the absolute value of the difference between the number of neighbors to the left and right of vi. The problem is also known as the Balanced Vertex Ordering problem and it finds many applications in graph drawing. We show that this problem is fixed parameter tractable and provide an algorithm that runs in time 2O(blogb)⋅nO(1). This resolves an open problem of Kára et al. [On the complexity of the balanced vertex ordering problem, in: COCOON, in: Lecture Notes in Comput. Sci., vol. 3595, 2005, pp. 849–858]. •The Imbalance Minimization problem on graphs asks for an ordering of the vertices such that the total imbalance is minimized.•The problem finds several applications in graph drawing, and is known to be NP-hard.•In this article, we show that the problem is fixed parameter tractable, when parameterized by the solution size, resolving an open question from the literature. In the Imbalance Minimization problem we are given a graph G=(V,E)G=(V,E) and an integer b and asked whether there is an ordering v sub(1)...v sub(n)v1...vn of V such that the sum of the imbalance of all the vertices is at most b . The imbalance of a vertex v sub(i)vi is the absolute value of the difference between the number of neighbors to the left and right of v sub(i)vi. The problem is also known as the Balanced Vertex Ordering problem and it finds many applications in graph drawing. We show that this problem is fixed parameter tractable and provide an algorithm that runs in time 2 super(O(blogb)) times n super(O(1))2O(blogb) times nO(1). This resolves an open problem of Kara et al. [On the complexity of the balanced vertex ordering problem, in: COCOON, in: Lecture Notes in Comput. Sci., vol. 3595, 2005, pp. 849-858]. In the Imbalance Minimization problem this paper is given a graph G=(V,E) and an integer b and asked whether there is an ordering v... ...v... of V such that the sum of the imbalance of all the vertices is at most b. The imbalance of a vertex v... is the absolute value of the difference between the number of neighbors to the left and right of v... The problem is also known as the Balanced Vertex Ordering problem and it finds many applications in graph drawing. This paper shows that this problem is fixed parameter tractable and provide an algorithm that runs in time 2...-n... This resolves an open problem of Kara et al. [On the complexity of the balanced vertex ordering problem, in: COCOON, in: Lecture Notes in Comput. Sci., vol. 3595, 2005, pp. 849-858]. (ProQuest: ... denotes formulae/symbols omitted.)
Author	Lokshtanov, Daniel Misra, Neeldhara Saurabh, Saket
Author_xml	– sequence: 1 givenname: Daniel surname: Lokshtanov fullname: Lokshtanov, Daniel email: daniello@ii.uib.no organization: University of California San Diego, San Diego, USA – sequence: 2 givenname: Neeldhara surname: Misra fullname: Misra, Neeldhara email: neeldhara@imsc.res.in organization: Institute of Mathematical Sciences, Chennai, India – sequence: 3 givenname: Saket surname: Saurabh fullname: Saurabh, Saket email: saket@imsc.res.in organization: Institute of Mathematical Sciences, Chennai, India
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SubjectTerms	Algorithms Balancing Computer science Data processing Fixed-parameter tractable algorithm Graph layout problems Graph theory Graphs Imbalance Integer programming Lectures Mathematical problems Numerical analysis Optimization Order disorder Parameterized complexity Studies
Title	Imbalance is fixed parameter tractable
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