A $(1+\varepsilon)$-Embedding of Low Highway Dimension Graphs into Bounded Treewidth Graphs

Feldmann, Andreas Emil; Fung, Wai Shing; Könemann, Jochen; Post, Ian

doi:10.1137/16m1067196

Cited by 19 publications

(33 citation statements)

References 13 publications

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“…for road networks. However, the definition of highway dimension relies on the notion of a hitting set of shortest path sets within network neighborhoods, and hence, e.g., exact computation of the parameter is known to be NP-hard even for unweighted networks [20]. This motivates us to look at other measures, which are both more locally defined and computationally tractable, while capturing essentially the same (or more) characteristics of the network's amenability to shortest path queries.…”

Section: Introductionmentioning

confidence: 99%

Beyond Highway Dimension: Small Distance Labels Using Tree Skeletons

Kosowski¹,

Viennot²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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The goal of a hub-based distance labeling scheme for a network G = (V, E) is to assign a small subset S(u) ⊆ V to each node u ∈ V , in such a way that for any pair of nodes u, v, the intersection of hub sets S(u) ∩ S(v) contains a node on the shortest uv-path.The existence of small hub sets, and consequently efficient shortest path processing algorithms, for road networks is an empirical observation. A theoretical explanation for this phenomenon was proposed by Abraham et al. (SODA 2010) through a network parameter they called highway dimension, which captures the size of a hitting set for a collection of shortest paths of length at least r intersecting a given ball of radius 2r. In this work, we revisit this explanation, introducing a more tractable (and directly comparable) parameter based solely on the structure of shortest-path spanning trees, which we call skeleton dimension. We show that skeleton dimension admits an intuitive definition for both directed and undirected graphs, provides a way of computing labels more efficiently than by using highway dimension, and leads to comparable or stronger theoretical bounds on hub set size.

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Section: Introductionmentioning

confidence: 99%

Beyond Highway Dimension: Small Distance Labels Using Tree Skeletons

Kosowski¹,

Viennot²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…Moreover, a related parameter is the highway dimension, which is used to model transportation networks. As shown by Feldmann et al [212] the techniques of Talwar [211] for low doubling metrics can be generalized to the highway dimension to obtain a QPTAS as well. Again, it is quite plausible to assume that a PAS exists.…”

Section: Network Designmentioning

confidence: 99%

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

et al. 2020

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“…Since the introduction of this graph parameter by Abraham et al [1,2], several papers try to match the performance of algorithms for low doubling dimension also in this setting. Building on Talwar's work [44], it was shown [22] that a QPTAS exists for problems such as Traveling Salesperson, Steiner Tree, Facility Location, and k-Median. For Bounded-Capacity Vehicle Routing a PTAS was shown [10], and the same work also gives approximation schemes for the k-Median and k-Center problems, when parameterizing by k and the highway dimension.…”

Section: A2 Further Related Workmentioning

confidence: 99%

Near-Linear Time Approximations Schemes for Clustering in Doubling Metrics

Saulpic

Cohen-Addad

Feldmann

2019

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

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We consider the classic Facility Location, k-Median, and k-Means problems in metric spaces of doubling dimension d. We give nearly linear-time approximation schemes for each problem. The complexity of our algorithms is 2 plogp1{εq{εq Opd 2 q n log 4 n`2 Opdq n log 9 n, making a significant improvement over the state-of-the-art algorithms which run in time n pd{εq Opdq .Moreover, we show how to extend the techniques used to get the first efficient approximation schemes for the problems of prize-collecting k-Medians and k-Means, and efficient bicriteria approximation schemes for k-Medians with outliers, k-Means with outliers and k-Center.help to handle some noise from the input: the k-Median objective can be dramatically perturbed by the addition of a few distant clients, which must then be discarded. Our resultsWe solve this open problem by proposing the first near-linear time algorithms for the k-Median and k-Means problems in metrics of fixed doubling dimension. More precisely, we show the following theorems, where we let f pεq " p1{εq 1{ε .Theorem 1.1. For any 0 ă ε ă 1{3, there exists a randomized p1`εq-approximation algorithm for k-Median in metrics of doubling dimension d with running time f pεq 2 Opd 2 q n log 4 n`2 Opdq n log 9 n and success probability at least 1´ε.Theorem 1.2. For any 0 ă ε ă 1{3, there exists a randomized p1`εq-approximation algorithm for k-Means in metrics of doubling dimension d with running time f pεq 2 Opd 2 q n log 5 n`2 Opdq n log 9 n and success probability at least 1´ε.Our results also extend to the Facility Location problem, in which no bound on the number of opened centers is given, but each center comes with an opening cost. The aim is to minimize the sum of the (1st power) of the distances from each point of the metric to its closest center, in addition to the total opening costs of all used centers.Theorem 1.3. For any 0 ă ε ă 1{3, there exists a randomized p1`εq-approximation algorithm for Facility Location in metrics of doubling dimension d with running time f pεq 2 Opd 2 q¨n`2 Opdq n log n and success probability at least 1´ε.In all these theorems, we make the common assumption to have access to the distances of the metric in constant time, as, e.g., in [18,27,29]. This assumption is discussed in Bartal et al. [9].Note that the double-exponential dependence on d is unavoidable unless P = NP, since the problems are APX-hard in Euclidean space of dimension d " Oplog nq. For Euclidean inputs, our algorithms for the k-Means and k-Median problems outperform the ones of Cohen-Addad [15], removing in particular the dependence on k, and the one of Kolliopoulos and Rao [32] when d ą 3, by removing the dependence on log d`6 n. Interestingly, for k " ωplog 9 nq our algorithm for the k-Means problem is faster than popular heuristics like k-Means++ which runs in time Opnkq in Euclidean space. We note that the success probability can be boosted to 1´ε δ by repeating the algorithm log δ times and outputting the best solution encountered.After proving the three theorems above, we will a...

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A $(1+\varepsilon)$-Embedding of Low Highway Dimension Graphs into Bounded Treewidth Graphs

Cited by 19 publications

References 13 publications

Beyond Highway Dimension: Small Distance Labels Using Tree Skeletons

Beyond Highway Dimension: Small Distance Labels Using Tree Skeletons

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Near-Linear Time Approximations Schemes for Clustering in Doubling Metrics

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