Approximation algorithms via contraction decomposition
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Foundations and Trends® in Theoretical Computer Science
Eigenvalue bounds, spectral partitioning, and metrical deformations via flows
Journal of the ACM (JACM)
Many sparse cuts via higher eigenvalues
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Structured recursive separator decompositions for planar graphs in linear time
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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In this paper, we address two long-standing questions about finding good separators in graphs of bounded genus and degree: 1. It is a classical result of Gilbert, Hutchinson, and Tarjan [J. Algorithms, 5 (1984), pp. 391-407] that one can find asymptotically optimal separators on these graphs if 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 $O(\sqrt{\smash[b]{g/n}})$ 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.