Approximation Algorithms for Multicommodity Flow and Normalized Cut Problems: Implementations and Experimental Study

Du, Ying; Chen, Danny Z.; Wu, Xiaodong

doi:10.1007/978-3-540-27798-9_14

Cited by 3 publications

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“…The normalized cut introduced by Shi and Malik [14] is useful in several areas. This measure is of interest not only for image segmentation but also for network theories and statistics [1,13,12,6,15].…”

Section: Introductionmentioning

confidence: 99%

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Bipartition of graphs based on the normalized cut and spectral methods

Perera,

Mizoguchi

2012

Preprint

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In the first part of this paper, we survey results that are associated with three types of Laplacian matrices:difference, normalized, and signless. We derive eigenvalue and eigenvector formulaes for paths and cycles using circulant matrices and present an alternative proof for finding eigenvalues of the adjacency matrix of paths and cycles using Chebyshev polynomials. Even though each results is separately well known, we unite them, and provide uniform proofs in a simple manner. The main objective of this study is to solve the problem of finding graphs, on which spectral clustering methods and normalized cuts produce different partitions. First, we derive a formula for a minimum normalized cut for graph classes such as paths, cycles, complete graphs, double-trees, cycle cross paths, and some complex graphs like lollipop graph LP n,m , roach type graph R n,k , and weighted path P n,k . Next, we provide characteristic polynomials of the normalized Laplacian matrices L(P n,k ) and L(R n,k ). Then, we present counter example graphs based on R n,k , on which spectral methods and normalized cuts produce different clusters.

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Section: Introductionmentioning

confidence: 99%

“…Graph G = (V, E) shown in the Figure3has vertex set V = {1, 2, 3, 4, 5, 6, 7} and edge set E = {(1, 2), (2, 3), (3, 1),(3, 4),(1, 4),(1, 5),(3,6),(6, 5),(7, 5),(7,6)}. Volume of the graph is 20.…”

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confidence: 99%