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Separators in Graphs

1998; Leighton, Rao 1999; Leighton, Rao

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Encyclopedia of Algorithms

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Keywords and Synonyms

Balanced cuts                    

Problem Definition

The (balanced) separator problem asks for a cut of minimum (edge)-weight in a graph, such that the two shores of the cut have approximately equal (node)-weight.

Formally, given an undirected graph \( G=(V,E) \), with a nonnegative edge-weight function \( c:E\to\mathbb{R}_+ \), a nonnegative node-weight function \( \pi:V\to\mathbb{R}_+ \), and a constant \( b\leq 1/2 \), a cut \( (S:V\setminus S) \) is said to be b -balanced, or a \( (b,1-b) \) -separator, if \( b\pi(V)\leq \pi(S)\leq (1-b)\pi(V) \) \( ( \)where \( \pi(S) \) stands for \( \sum_{v\in S} \pi(v) \) \( ) \).

Problem 1 (b-balanced separator)

Input: Edge- and node-weighted graph \( G=(V,E,c,\pi) \), constant \( b\leq 1/2 \).

Output: A b-balanced cut \( (S:V\setminus S) \). Goal: minimize the edge weight \( c(\delta(S)) \).

Closely related is the product sparsest cut problem.

Problem 2 ((Product) Sparsest cut)

Input: Edge- and node-weighted graph \( G=(V,E,c,\pi) \).

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Arizona State University, Tempe, AZ, USA

    Goran Konjevod

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  1. Goran Konjevod
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Editors and Affiliations

  1. Department of Electrical Engineering and Computer ScienceMcCormick School of Engineering and Applied Science, Northwestern University, Evanston, IL, 60208, USA

    Ming-Yang Kao Professor of Computer Science

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Konjevod, G. (2008). Separators in Graphs. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_362

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