A separator theorem for graphs of bounded genus
Journal of Algorithms
Asymptotic theory of finite dimensional normed spaces
Asymptotic theory of finite dimensional normed spaces
Excluded minors, network decomposition, and multicommodity flow
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Rounding via trees: deterministic approximation algorithms for group Steiner trees and k-median
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
On approximating arbitrary metrices by tree metrics
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Small distortion and volume preserving embeddings for planar and Euclidean metrics
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
Placement algorithms for hierarchical cooperative caching
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Multicommodity max-flow min-cut theorems and their use in designing approximation algorithms
Journal of the ACM (JACM)
Approximation algorithms for the 0-extension problem
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
An improved approximation algorithm for the 0-extension problem
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Optimal System of Loops on an Orientable Surface
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
The intrinsic dimensionality of graphs
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
A tight bound on approximating arbitrary metrics by tree metrics
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
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
Bounded Geometries, Fractals, and Low-Distortion Embeddings
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
Approximate classification via earthmover metrics
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Expander flows, geometric embeddings and graph partitioning
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Bypassing the embedding: algorithms for low dimensional metrics
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Improved approximation algorithms for minimum-weight vertex separators
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
On distance scales, embeddings, and efficient relaxations of the cut cone
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Embeddings of negative-type metrics and an improved approximation to generalized sparsest cut
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
On hierarchical routing in doubling metrics
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Greedy optimal homotopy and homology generators
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
How to Play Unique Games Using Embeddings
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Strong-diameter decompositions of minor free graphs
Proceedings of the nineteenth annual ACM symposium on Parallel algorithms and architectures
Eigenvalue Bounds, Spectral Partitioning, and Metrical Deformations via Flows
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Volume Distortion for Subsets of Euclidean Spaces
Discrete & Computational Geometry
Metric Embeddings with Relaxed Guarantees
SIAM Journal on Computing
Minimum cuts and shortest non-separating cycles via homology covers
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Multi-way spectral partitioning and higher-order cheeger inequalities
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Maximum edge-disjoint paths in k-sums of graphs
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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We study the quantitative geometry of graphs in terms of their genus, using the structure of certain "cut graphs," i.e. subgraphs whose removal leaves a planar graph. In particular, we give optimal bounds for random partitioning schemes, as well as various types of embeddings. Using these geometric primitives, we present exponentially improved dependence on genus for a number of problems like approximate max-flow/min-cut theorems, approximations for uniform and nonuniform Sparsest Cut, treewidth approximation, Laplacian eigenvalue bounds, and Lipschitz extension theorems and related metric labeling problems. We list here a sample of these improvements. All the following statements refer to graphs of genus g, unless otherwise noted. • We show that such graphs admit an O(log g)-approximate multi-commodity max-flow/min-cut theorem for the case of uniform demands. This bound is optimal, and improves over the previous bound of O(g) [KPR93, FT03]. For general demands, we show that the worst possible gap is O(log g + CP), where CP is the gap for planar graphs. This dependence is optimal, and already yields a bound of O(log g + √log n), improving over the previous bound of O(√g log n) [KLMN04]. • We give an O(√log g)-approximation for the uniform Sparsest Cut, balanced vertex separator, and treewidth problems, improving over the previous bound of O(g) [FHL05]. • If a graph G has genus g and maximum degree D, we show that the kth Laplacian eigenvalue of G is (log g)2 · O(kgD/n), improving over the previous bound of g2·O(kgD/n) [KLPT09]. There is a lower bound of Ω(kgD/n), making this result almost tight. • We show that if (X, d) is the shortest-path metric on a graph of genus g and S ⊆ X, then every L-Lipschitz map f: S → Z into a Banach space Z admits an O(L log g)-Lipschitz extension f: X → Z. This improves over the previous bound of O(Lg) [LN05], and compares to a lower bound of Ω(L√log g). In a related way, we show that there is an O(log g)-approximation for the 0-extension problem on such graphs, improving over the previous O(g) bound. • We show that every n-vertex shortest-path metric on a graph of genus g embeds into L2 with distortion O(log g + √log n), improving over the previous bound of O(√g log n). Our result is asymptotically optimal for every dependence g = g(n).