A separator theorem for graphs of bounded genus
Journal of Algorithms
Separators in two and three dimensions
STOC '90 Proceedings of the twenty-second annual ACM symposium on Theory of computing
Spectral k-way ratio-cut partitioning and clustering
Proceedings of the 1993 symposium on Research on integrated systems
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STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Recent directions in netlist partitioning: a survey
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Journal of the ACM (JACM)
Geometric Separators for Finite-Element Meshes
SIAM Journal on Scientific Computing
Combinatorial aspects of geometric graphs
Computational Geometry: Theory and Applications
On the Quality of Spectral Separators
SIAM Journal on Matrix Analysis and Applications
Small distortion and volume preserving embeddings for planar and Euclidean metrics
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Shallow excluded minors and improved graph decompositions
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Normalized Cuts and Image Segmentation
IEEE Transactions on Pattern Analysis and Machine Intelligence
Approximation algorithms for the 0-extension problem
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
On average distortion of embedding metrics into the line and into L1
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Probabilistic approximation of metric spaces and its algorithmic applications
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Convex Optimization
Improved approximation algorithms for minimum-weight vertex separators
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Spectral Partitioning, Eigenvalue Bounds, and Circle Packings for Graphs of Bounded Genus
SIAM Journal on Computing
Conductance and convergence of Markov chains-a combinatorial treatment of expanders
SFCS '89 Proceedings of the 30th Annual Symposium on Foundations of Computer Science
Many sparse cuts via higher eigenvalues
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Proceedings of the forty-fifth 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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We present a new method for upper bounding the second eigenvalue of the Laplacian of graphs. Our approach uses multi-commodity flows to deform the geometry of the graph; we embed the resulting metric into Euclidean space to recover a bound on the Rayleigh quotient. Using this, we show that every n-vertex graph of genus g and maximum degree D satisfies λ2(G)=O((g+1)3D/n). This recovers the O(D/n) bound of Spielman and Teng for planar graphs, and compares to Kelner's bound of O((g+1)poly(D)/n), but our proof does not make use of conformal mappings or circle packings. We are thus able to extend this to resolve positively a conjecture of Spielman and Teng, by proving that λ2(G) = O(Dh6log h/n) whenever G is Kh-minor free. This shows, in particular, that spectral partitioning can be used to recover O(&sqrt;n)-sized separators in bounded degree graphs that exclude a fixed minor. We extend this further by obtaining nearly optimal bounds on λ2 for graphs that exclude small-depth minors in the sense of Plotkin, Rao, and Smith. Consequently, we show that spectral algorithms find separators of sublinear size in a general class of geometric graphs. Moreover, while the standard “sweep” algorithm applied to the second eigenvector may fail to find good quotient cuts in graphs of unbounded degree, our approach produces a vector that works for arbitrary graphs. This yields an alternate proof of the well-known nonplanar separator theorem of Alon, Seymour, and Thomas that states that every excluded-minor family of graphs has O(&sqrt;n)-node balanced separators.