Advances in Metric Ramsey Theory and its Applications

Bartal, Yair

doi:10.48550/arxiv.2104.03484

Cited by 4 publications

(7 citation statements)

References 23 publications

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“…Clan embeddings were also studied for trees [FL21]. Clan embeddings are somewhat similar to the previously introduced multiembeddings [BM04,Bar21] in that each vertex is mapped to a subset of vertices. However, multiembeddings lack the notion of chief, which is crucial in all our applications.…”

Section: Related Workmentioning

confidence: 97%

Low Treewidth Embeddings of Planar and Minor-Free Metrics

Filtser¹,

Lê²

2022

Preprint

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Cohen-Addad, Filtser, Klein and Le [FOCS'20] constructed a stochastic embedding of minorfree graphs of diameter D into graphs of treewidth O (log n) with expected additive distortion + D. Cohen-Addad et al. then used the embedding to design the first quasi-polynomial time approximation scheme (QPTAS) for the capacitated vehicle routing problem. Filtser and Le [STOC'21] used the embedding (in a different way) to design a QPTAS for the metric Baker's problems in minor-free graphs. In this work, we devise a new embedding technique to improve the treewidth bound of Cohen-Addad et al. exponentially to O (log log n) 2 . As a corollary, we obtain the first efficient PTAS for the capacitated vehicle routing problem in minor-free graphs. We also significantly improve the running time of the QPTAS for the metric Baker's problems in minor-free graphs from n O (log(n)) to n O (log log(n)) 3 . Applying our embedding technique to planar graphs, we obtain a deterministic embedding of planar graphs of diameter D into graphs of treewidth O((log log n) 2 ) ) and additive distortion + D that can be constructed in nearly linear time. Important corollaries of our result include a bicriteria PTAS for metric Baker's problems and a PTAS for the vehicle routing problem with bounded capacity in planar graphs, both run in almost-linear time. The running time of our algorithms is significantly better than previous algorithms that require quadratic time.A key idea in our embedding is the construction of an (exact) emulator for tree metrics with treewidth O(log log n) and hop-diameter O(log log n). This result may be of independent interest.

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Section: Related Workmentioning

confidence: 97%

Low Treewidth Embeddings of Planar and Minor-Free Metrics

Filtser¹,

Lê²

2022

Preprint

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“…A key tool we use is Bartal's Ramsey decomposition lemma [3]. Since we will use it in slightly more general form than the original, we rephrase and reprove it here.…”

Section: Bartal's Ramsey Decomposition Lemmamentioning

confidence: 99%

“…The construction of the ultrametric skeleton here follows the construction in [11], which in turn uses Bartal's Ramsey decomposition lemma [3] as a key tool.…”

Section: Introductionmentioning

confidence: 99%

Dvoretzky-type theorem for Ahlfors regular spaces

Mendel¹

2021

Preprint

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It is proved that for any 0 < β < α, any bounded Ahlfors α-regular space contains a β-regular compact subset that embeds biLipschitzly in an ultrametric with distortion at most O(α/(α − β)). The bound on the distortion is asymptotically tight when β → α. The main tool used in the proof is a regular form of the ultrametric skeleton theorem.Here C c > 0 are independent of x and r. Ahlfors α-regular space X has, in particular, Hausdorff dimension dim H (X) = α. For more information on Ahlfors regular spaces and their importance, see [8, 5,7].An ultrametric space is a metric space (U, ρ) satisfying the strengthened triangle inequality ρ(x, y) max{ρ(x, z), ρ(y, z)} for all x, y, z ∈ U. Saying that (X, d) embeds (biLipschitzly) with distortion D ∈ [1, ∞) into an ultrametric space means that there exists an ultrametric ρ on X satisfying d(x, y) ρ(x, y) Dd(x, y) for all x, y ∈ X. The ultrametric distortion of X is the infimum over D for which X embeds in an ultrametric with distortion at most D.In this paper we study regular (approximate) ultrametric subsets of Ahlfors regular spaces. Arcozzi et. al. [1, Theorem 1] proved that for every 0 < β < α, any bounded Ahlfors α-regular

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“…The idea of one-to-many embedding of graphs was originated by Bartal and Mendel [BM04], who for k ≥ 1 constructed embedding into ultrametric with O(n 1+ 1 k ) nodes and path distortion O(k⋅log n⋅ log log n) (see Definition 8, and ignore all hop constrains). The path distortion was later improved to O(k ⋅ log n) [FL21a,Bar21]. Recently, Haeupler et al [HHZ21a] studied approximate copy tree embedding which is essentially equivalent to one-to-many tree embeddings.…”

Section: Related Workmentioning

confidence: 99%

Hop-Constrained Metric Embeddings and their Applications

Filtser¹

2021

Preprint

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In network design problems, such as compact routing, the goal is to route packets between nodes using the (approximated) shortest paths. A desirable property of these routes is a small number of hops, which makes them more reliable, and reduces the transmission costs.Following the overwhelming success of stochastic tree embeddings for algorithmic design, Haeupler, Hershkowitz, and Zuzic (STOC'21) studied hop-constrained Ramsey-type metric embeddings into trees. Specifically, embedding and d(h) G (u, v) denotes the minimum weight of a u − v path with at most h hops. Haeupler et al. constructed embedding where M contains 1 − fraction of the vertices and β = t = O( log 2 n). They used their embedding to obtain multiple bicriteria approximation algorithms for hop-constrained network design problems.In this paper, we first improve the Ramsey-type embedding to obtain parameters t = β = Õ(log n) , and generalize it to arbitrary distortion parameter t (in the cost of reducing the size of * The research was supported by the Simons Foundation.

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Advances in Metric Ramsey Theory and its Applications

Cited by 4 publications

References 23 publications

Low Treewidth Embeddings of Planar and Minor-Free Metrics

Low Treewidth Embeddings of Planar and Minor-Free Metrics

Dvoretzky-type theorem for Ahlfors regular spaces

Hop-Constrained Metric Embeddings and their Applications

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