An Improved Decomposition Theorem for Graphs Excluding a Fixed Minor

Fakcharoenphol, Jittat; Talwar, Kunal

doi:10.1007/978-3-540-45198-3_4

“…Additionally, if X is the shortest-path metric on an n-point constant-degree expander, then α X = Ω(log n) [4]. For every metric space X induced by an edge-weighted graph which excludes K r,r as a minor, it is shown in the sequence of papers [19,33,11] …”

Section: Definition 12 (Decomposition Bundle) Given a Functionmentioning

confidence: 99%

“…The first one exhibits a family of decompositions with respect to a diameter bound ∆ > 0; it follows easily from [19], with improved constants due to [11]. Note that in contrast to Definition 1.1 (and also to Rao's embedding [33]), we require that x and y are padded simultaneously.…”

Section: Optimal Volume-respecting Embeddingsmentioning

confidence: 99%

Measured descent: a new embedding method for finite metrics

Krauthgamer

¹

,

Lee

²

,

Mendel

³

et al. 2005

GAFA, Geom. funct. anal.

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We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Fréchet embeddings for finite metrics, due to [Bourgain, 1985] and [Rao, 1999]. We prove that any n-point metric space (X, d) embeds in Hilbert space with distortion O( √ α X · log n), where α X is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an O( (log λ X ) log n) distortion embedding, where λ X is the doubling constant of X. Since λ X ≤ n, this result recovers Bourgain's theorem, but when the metric X is, in a sense, "low-dimensional," improved bounds are achieved.Our embeddings are volume-respecting for subsets of arbitrary size. One consequence is the existence of (k, O(log n)) volume-respecting embeddings for all 1 ≤ k ≤ n, which is the best possible, and answers positively a question posed by U. Feige. Our techniques are also used to answer positively a question of Y. Rabinovich, showing that any weighted n-point planar graph embeds in ℓ O(log n) ∞ with O(1) distortion. The O(log n) bound on the dimension is optimal, and improves upon the previously known bound of O((log n) 2 ).

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“…We note that the previous best bounds were due to Feige [12], who showed that a variant of Bourgain's embedding achieves distortion O( √ log n · √ log n + k log k) (note that this is Ω( √ n) for large values of k), and to Rao who showed that O((log n) 3/2 ) volume distortion is achievable for all 1 ≤ k ≤ n (this follow indirectly from [33], and was first observed in [13]). This also improves the dependence on r in Rao's volume-respecting embeddings of K r,r -excluded metrics, from (k, O(r 2 √ log n)), due to [33,11], to (k, O(r √ log n)). 1 As a corollary, we obtain an improved O(r √ log n)-approximate max-flow/min-cut algorithm for graphs which exclude K r,r as a minor.…”

Section: Resultsmentioning

confidence: 83%

“…Continuing this way a total of s times, we end up with 3 s partitions, and in at least one of them, neither B(x, δ) nor B(y, δ) is cut. The results of [19,11] show there exists a constant c > 0 such that the diameter of each cluster in the resulting partitions is at most cs 2 δ, and the lemma follows by setting δ = ∆/(cs 2 ).…”

Section: Optimal Volume-respecting Embeddingsmentioning

confidence: 92%

Measured Descent: A New Embedding Method for Finite Metrics

Krauthgamer

¹

,

Lee

²

,

Mendel

³

et al.

45th Annual IEEE Symposium on Foundations of Computer Science

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We devise a new embedding technique, which we call measured descent, based on decomposing a metric space locally, at varying speeds, according to the density of some probability measure. This provides a refined and unified framework for the two primary methods of constructing Fréchet embeddings for finite metrics, due to [Bourgain, 1985] and [Rao, 1999]. We prove that any n-point metric space (X, d) embeds in Hilbert space with distortion O( √ α X · log n), where α X is a geometric estimate on the decomposability of X. As an immediate corollary, we obtain an O( (log λ X ) log n) distortion embedding, where λ X is the doubling constant of X. Since λ X ≤ n, this result recovers Bourgain's theorem, but when the metric X is, in a sense, "low-dimensional," improved bounds are achieved.Our embeddings are volume-respecting for subsets of arbitrary size. One consequence is the existence of (k, O(log n)) volume-respecting embeddings for all 1 ≤ k ≤ n, which is the best possible, and answers positively a question posed by U. Feige. Our techniques are also used to answer positively a question of Y. Rabinovich, showing that any weighted n-point planar graph embeds in ℓ O(log n) ∞ with O(1) distortion. The O(log n) bound on the dimension is optimal, and improves upon the previously known bound of O((log n)2 ).

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“…In graphs excluding a K r minor, they show a gap of O(r 3 ) which was improved to O(r 2 ) in [11]. Theorem 1.3 and Corollary 1.4 show that we can obtain a constant gap while ensuring that the flow for each pair is along a single path.…”

Section: Corollary 14 the Maxflow-mincut Gap For Product Multicommodmentioning

confidence: 97%

Edge-Disjoint Paths in Planar Graphs

Chekuri¹,

Khanna

²

,

Shepherd³

45th Annual IEEE Symposium on Foundations of Computer Science

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We study the maximum edge-disjoint paths problem (MEDP). We are given a graph G = (V,E) and a set Τ = {s 1 t 1 , s 2 t 2 , . . . , s k t k } of pairs of vertices: the objective is to find the maximum number of pairs in Τ that can be connected via edge-disjoint paths. Our main result is a poly-logarithmic approximation for MEDP on undirected planar graphs if a congestion of 2 is allowed, that is, we allow up to 2 paths to share an edge. Prior to our work, for any constant congestion, only a polynomial-factor approximation was known for planar graphs although much stronger results are known for some special cases such as grids and grid-like graphs. We note that the natural multicommodity flow relaxation of the problem has an integrality gap of Ω(√|V|) even on planar graphs when no congestion is allowed. Our starting point is the same relaxation and our result implies that the integrality gap shrinks to a poly-logarithmic factor once 2 paths are allowed per edge. Our result also extends to the unsplittable flow problem and the maximum integer multicommodity flow problem.A set X ⊆ V is well-linked if for each S ⊂ V , |δ(S)| ≥ min{|S ∩ X|, |(V -S) ∩ X|}. The heart of our approach is to show that in any undirected planar graph, given any matching M on a well-linked set X, we can route Ω(|M|) pairs in M with a congestion of 2. Moreover, all pairs in M can be routed with constant congestion for a sufficiently large constant. This results also yields a different proof of a theorem of Klein, Plotkin, and Rao that shows an O(1) maxflow-mincut gap for uniform multicommodity flow instances in planar graphs.The framework developed in this paper applies to general graphs as well. If a certain graph theoretic conjecture is true, it will yield poly-logarithmic integrality gap for MEDP with constant congestion. This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This conference paper is available at ScholarlyCommons: http://repository.upenn.edu/cis_papers/64 Edge-Disjoint Paths in Planar Graphs

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An Improved Decomposition Theorem for Graphs Excluding a Fixed Minor

Cited by 56 publications

References 25 publications

Measured descent: a new embedding method for finite metrics

Measured descent: a new embedding method for finite metrics

Measured Descent: A New Embedding Method for Finite Metrics

Edge-Disjoint Paths in Planar Graphs

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