On Light Spanners, Low-treewidth Embeddings and Efficient Traversing in Minor-free Graphs

Cohen-Addad, Vincent; Filtser, Arnold; Klein, Philip N.; Le, Hung

doi:10.48550/arxiv.2009.05039

Cited by 2 publications

(15 citation statements)

References 78 publications

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“…Alternately, one can try the metric embedding approach (for which bicriteria approximation is inherit). Unfortunately, unlike the classic embeddings in [EKM14, FKS19], Cohen-Addad et al [CFKL20] provided a stochastic embedding with expected distortion gurantee. Such a stochastic gurantee is not strong enough to construct approximation schemes for the metric ρ-independent/dominating set problems.…”

Section: Metric Becker Problems In Minor Free Graphsmentioning

confidence: 99%

“…Following the success in planar graphs, Cohen-Addad et al [CFKL20] wanted to generalize to minor free graphs. Unfortunately, they showed that already obtaining additive distortion 1 20 D for K 6 -free graphs requires embedding into treewidth Ω( √ n) graphs.…”

Section: Introductionmentioning

confidence: 99%

“…Unfortunately, they showed that already obtaining additive distortion 1 20 D for K 6 -free graphs requires embedding into treewidth Ω( √ n) graphs. Inspired by the case of trees, [CFKL20] bypass this barrier by constructing a stochastic embedding from K r -free n-vertex graphs into distribution over treewidth O r ( log n…”

Section: Introductionmentioning

confidence: 99%

“…Similarly to the case in planar graphs, Cohen-Addad et al [CFKL20] used their embedding to construct an approximation scheme for the capacitated vehicle routing problem in K r -minor-free graphs. However, due to the stochastic nature of the embedding, it was not strong enough to imply any results for the metric ρ-dominating/isolated problems in minor free graphs, which remain wide open.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, similarly to the case of trees, we construct Ramsey type and clan embedding analogs to the stochastic embedding of [CFKL20]. Our Ramsey type embedding bypasses the lower bound of Ω( √ n) from [CFKL20] while guaranteeing worst case distortion (for a large random subset of vertices).…”

Section: Introductionmentioning

confidence: 99%

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Clan Embeddings into Trees, and Low Treewidth Graphs

Filtser

2021

Preprint

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In low distortion metric embeddings, the goal is to embed a host "hard" metric space into a "simpler" target space, while approximately preserving pairwise distances. A highly desirable target space is that of a tree metric. Unfortunately, such embedding will result in a huge distortion. A celebrated bypass to this problem is stochastic embedding with logarithmic expected distortion. Another bypass is Ramsey type embedding, where the distortion guarantee applies only to a subset of the points. However both this solutions fail to provide an embedding into a single tree with worst case distortion guarantee on all pairs. In this paper we propose a novel third bypass called clan embedding. Here each point x is mapped to a subset of points f (x) (called a clan) with a special chief point χ(x) ∈ f (x). The clan embedding has multiplicative distortion t if for every pair x, y some copy y ′ ∈ f (y) in the clan of y is close to the chief of x: min y ′ ∈f (y) d(y ′ , χ(x)) ≤ t ⋅ d(x, y). Our first result is clan embedding into a tree with multiplicative distortion O( log n ) such that each point has 1 + copies (in expectation). In addition, for graphs we provide a "spanning" version of this theorem, and use it to devise the first compact routing scheme with constant size routing tables.Next we turn to minor free graphs, who were previously stochastically embedded into bounded treewidth graphs with expected additive distortion D (D being the diameter). We devise a Ramsey type embedding and clan embedding analogs of the stochastic embedding. We use this embeddings to construct the first (bicriteria quasi-polynomial) approximation schemes for the metric ρ-dominating set and metric ρ-independent set problems in minor free graphs.

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Section: Metric Becker Problems In Minor Free Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Clan Embeddings into Trees, and Low Treewidth Graphs

Filtser

2021

Preprint

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Hop-Constrained Metric Embeddings and their Applications

Filtser

2022

2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)

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Given a metric space (X, d X ), a (β, s, ∆)-sparse cover is a collection of clusters C ⊆ P (X) with diameter at most ∆, such that for every point x ∈ X, the ball B X (x, ∆ β ) is fully contained in some cluster C ∈ C, and x belongs to at most s clusters in C. Our main contribution is to show that the shortest path metric of every K r -minor free graphs admits (O(r), O(r 2 ), ∆)sparse cover, and for every ϵ > 0, (4 + ϵ, O( 1 ϵ ) r , ∆)-sparse cover (for arbitrary ∆ > 0). We then use this sparse cover to show that every K r -minor free graph embeds into ℓ)⋅log n ∞ with distortion O(r)). Further, we provide applications of these sparse covers into padded decompositions, sparse partitions, universal TSP / Steiner tree, oblivious buy at bulk, name independent routing, and path reporting distance oracles.

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On Light Spanners, Low-treewidth Embeddings and Efficient Traversing in Minor-free Graphs

Cited by 2 publications

References 78 publications

Clan Embeddings into Trees, and Low Treewidth Graphs

Clan Embeddings into Trees, and Low Treewidth Graphs

Hop-Constrained Metric Embeddings and their Applications

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