Minor-Free Graphs Have Light Spanners

Borradaile, Glencora; Le, Hung; Wulff-Nilsen, Christian

doi:10.1109/focs.2017.76

Cited by 26 publications

(63 citation statements)

References 22 publications

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“…There are some minor details specific to our presentation of the proof, mostly involving the calculation of diameter upper bounds because our clusters are constructed via K, whose edges do not belong to S i . In retrospect, both papers of Borradaile, Le and Wulff-Nilsen [15,14] implicitly proved Lemma 6 but technical details specific to their problems obfuscate a clean statement. By phrasing their techniques in a single lemma, we believe that it would be of independent interest.…”

Section: Bounding the Spanner Weightmentioning

confidence: 99%

“…These constructions often require another property of the input such as the packing property, that we don not have in our setting. The light spanner construction by Borradaile, Le and Wulff-Nilsen [14] for H-minor-free graphs uses the fact that a contracted graph of an H-minor-free graph has a linear number of edges. However, if one applies this construction to terminals, the contracted graph may have a super-linear number of edges w.r.t the number of terminals.…”

Section: Our Techniquesmentioning

confidence: 99%

“…In [15], the packing property of doubling metrics was used to argue that K has bounded degree (see Lemma 3.1 in [15]), so they can afford to buy every edge of K to the spanner using -clusters' credits. Problems considered in prior work [14,15,42] enjoy the same degree boundedness or average-degree boundedness. In our setting, we do not have any constraint on the degree of K; it could be a complete graph.…”

Section: Sparse Spanner Oracles Imply Light Subset Spannersmentioning

confidence: 99%

“…The weak sparsity of O guarantees that on average, each high degree node of K only pays for the weight equal to the total weight of a constant number of edges of K. Intuitively, sparse spanner oracles allow us to "reduce the degree" of K to constant. The rest of the argument can therefore be adapted directly from prior work [14,15,42]. In the following section, we give a formal argument.…”

Section: Sparse Spanner Oracles Imply Light Subset Spannersmentioning

confidence: 99%

“…Over the past 20 years, much research have been spent on exploiting minor-closed properties to design polynomial time approximation schemes 1 (PTAS) for TSP and subset TSP [29,4,37,38,13,24,14,30]. In FOCS'17, Borradaile, Le and Wulff-Nilsen [14] fully resolved the approximation complexity of TSP in H-minor-free graphs for any fixed graph H by designing an efficient PTAS (EPTAS). However, designing a PTAS, even an inefficient one, for subset TSP in the same setting remains a widely open problem that has been raised several times [24,13,14].…”

Section: Introductionmentioning

confidence: 99%

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A PTAS for subset TSP in minor-free graphs

Le¹

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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We give the first PTAS for the subset Traveling Salesperson Problem (TSP) in H-minorfree graphs. This resolves a long standing open problem in a long line of work on designing PTASes for TSP in minor-closed families initiated by Grigni, Koutsoupias and Papadimitriou in FOCS'95. The main technical ingredient in our PTAS is a construction of a nearly light subset (1 + )-spanner for any given edge-weighted H-minor-free graph. This construction is based on a necessary and sufficient condition given by sparse spanner oracles: light subset spanners exist if and only if sparse spanner oracles exist. This relationship allows us to obtain two new results: • An (1 + )-spanner with lightness O( −d+2 ) for any doubling metric of constant dimension d. This improves the earlier lightness bound −O(d) obtained by Borradaile, Le and Wulff-Nilsen [15].

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Section: Bounding the Spanner Weightmentioning

confidence: 99%

Section: Our Techniquesmentioning

confidence: 99%

Section: Sparse Spanner Oracles Imply Light Subset Spannersmentioning

confidence: 99%

Section: Sparse Spanner Oracles Imply Light Subset Spannersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A PTAS for subset TSP in minor-free graphs

Le¹

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Light Spanners for High Dimensional Norms via Stochastic Decompositions

Filtser

Neiman

2022

Algorithmica

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Spanners for low dimensional spaces (e.g. Euclidean space of constant dimension, or doubling metrics) are well understood. This lies in contrast to the situation in high dimensional spaces, where except for the work of Har-Peled, Indyk and Sidiropoulos (SODA 2013), who showed that any n-point Euclidean metric has an O(t)-spanner with Õ(n 1+1/t 2 ) edges, little is known.In this paper we study several aspects of spanners in high dimensional normed spaces. First, we build spanners for finite subsets of p with 1 < p ≤ 2. Second, our construction yields a spanner which is both sparse and also light, i.e., its total weight is not much larger than that of the minimum spanning tree. In particular, we show that any n-point subset of p for 1 < p ≤ 2 has an O(t)-spanner with n 1+ Õ(1/t p ) edges and lightness n Õ(1/t p ) .In fact, our results are more general, and they apply to any metric space admitting a certain low diameter stochastic decomposition. It is known that arbitrary metric spaces have an O(t)spanner with lightness O(n 1/t ). We exhibit the following tradeoff: metrics with decomposability parameter ν = ν(t) admit an O(t)-spanner with lightness Õ(ν 1/t ). For example, n-point Euclidean metrics have ν ≤ n 1/t , metrics with doubling constant λ have ν ≤ λ, and graphs of genus g have ν ≤ g. While these families do admit a (1 + )-spanner, its lightness depend exponentially on the dimension (resp. log g). Our construction alleviates this exponential dependency, at the cost of incurring larger stretch.

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Truly Optimal Euclidean Spanners

Solomon

2019

2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)

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Euclidean spanners are important geometric structures, having found numerous applications over the years. Cornerstone results in this area from the late 80s and early 90s state that for any ddimensional n-point Euclidean space, there exists a (1 + )-spanner with n · O( −d+1 ) edges and lightness (normalized weight) O( −2d ). 1 Surprisingly, the fundamental question of whether or not these dependencies on and d for small d can be improved has remained elusive, even for d = 2. This question naturally arises in any application of Euclidean spanners where precision is a necessity (thus is tiny). In the most extreme case is inverse polynomial in n, and then one could potentially improve the size and lightness bounds by factors that are polynomial in n.The state-of-the-art bounds n · O( −d+1 ) and O( −2d ) on the size and lightness of spanners are realized by the greedy spanner. In 2016, Filtser and Solomon [25] proved that, in low dimensional spaces, the greedy spanner is "near-optimal"; informally, their result states that the greedy spanner for dimension d is just as sparse and light as any other spanner but for dimension larger by a constant factor. Hence the question of whether the greedy spanner is truly optimal remained open to date.The contribution of this paper is two-fold.1 Introduction Background and motivationSparse spanners. Let P be a set of n points in R d , d ≥ 2, and consider the complete weighted graph G P = (P, P 2 ) induced by P , where the weight of any edge (x, y) ∈ P 2 is the Euclidean distance |xy| between its endpoints. Let H = (P, E) be a spanning subgraph of G P , with E ⊆ P 2 , where, as in G P , the weight function is given by the Euclidean distances. For any t ≥ 1, H is called a t-spanner for P if for every x, y ∈ P , the distance d G (x, y) between x and y in G is at most t|xy|; the parameter t is called the stretch of the spanner and the most basic goal is to get it down to 1 + , for arbitrarily small > 0, 1 The lightness of a spanner is the ratio of its weight and the MST weight. without using too many edges. Euclidean spanners were introduced in the pioneering SoCG'86 paper of Chew [16], who showed that O(n) edges can be achieved with stretch √ 10, and later improved the stretch bound to 2 [17]. The first Euclidean spanners with stretch 1 + , for an arbitrarily small > 0, were presented independently in the seminal works of Clarkson [18] (FOCS'87) and Keil [38] (see also [39]), which introduced the Θ-graph in R 2 and R 3 , and soon afterwards was generalized for any R d in [45,2]. The Θ-graph is a natural variant of the Yao graph, introduced by Yao [51] in 1982, where, roughly speaking, the space R d around each point p ∈ P is partitioned into cones of angle Θ each, and then edges are added between each point p ∈ P and its closest points in each of the cones centered around it. The Θ-graph is defined similarly, where, instead of connecting p to its closest point in each cone, we connect it to a point whose orthogonal projection to some fixed ray contained in the cone is closest to p....

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Minor-Free Graphs Have Light Spanners

Cited by 26 publications

References 22 publications

A PTAS for subset TSP in minor-free graphs

A PTAS for subset TSP in minor-free graphs

Light Spanners for High Dimensional Norms via Stochastic Decompositions

Truly Optimal Euclidean Spanners

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