Spectral partitioning, eigenvalue bounds, and circle packings for graphs of bounded genus
Name
60678254-MIT.pdf
Description
Full printable version
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1.76 MB
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620efb4338522468cf1d836889cb4fb4
Author(s)
Kelner, Jonathan, 1980-
Advisor(s)
Daniel Spielman.
Date Issued
2005
Publisher
Massachusetts Institute of Technology
Abstract
In this paper, we address two longstanding questions about finding good separators in graphs of bounded genus and degree: 1. It is a classical result of Gilbert, Hutchinson, and Tarjan [12] that one can find asymptotically optimal separators on these graphs if he is given both the graph and an embedding of it onto a low genus surface. Does there exist a simple, efficient algorithm to find these separators given only the graph and not the embedding? 2. In practice, spectral partitioning heuristics work extremely well on these graphs. Is there a theoretical reason why this should be the case? We resolve these two questions by showing that a simple spectral algorithm finds separators of cut ratio O(sqrt(g/n)) and vertex bisectors of size O(sqrt(gn)) in these graphs, both of which are optimal. As our main technical lemma, we prove an O(g/n) bound on the second smallest eigenvalue of the Laplacian of such graphs and show that this is tight, thereby resolving a conjecture of Spielman and Teng. While this lemma is essentially combinatorial in nature, its proof comes from continuous mathematics, drawing on the theory of circle packings and the geometry of compact Riemann surfaces.
Description
Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.
Includes bibliographical references (leaves 37-39).
Subjects
Electrical Engineering and Computer Science.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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