Travelling on Graphs with Small Highway Dimension

Disser, Yann; Feldmann, Andreas; Klimm, Max; Könemann, Jochen

doi:10.1007/978-3-030-30786-8_14

Cited by 3 publications

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“…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. The k-Center problem is hard even in the parameterized setting [23,24], and Traveling Salesperson was recently shown [20] to be weakly NP-hard even if the highway dimension is 1. It is an intriguing question whether the problems studied in this paper also admit PTASs for low highway dimension graphs.…”

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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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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Travelling on Graphs with Small Highway Dimension

et al. 2021

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Polynomial time approximation schemes for clustering in low highway dimension graphs

Feldmann

Saulpic²

2021

Journal of Computer and System Sciences

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We study clustering problems such as k-Median, k-Means, and Facility Location in graphs of low highway dimension, which is a graph parameter modeling transportation networks. It was previously shown that approximation schemes for these problems exist, which either run in quasi-polynomial time (assuming constant highway dimension) [Feldmann et al. SICOMP 2018] or run in FPT time (parameterized by the number of clusters k, the highway dimension, and the approximation factor) [Becker et al. ESA 2018, Braverman et al. 2020. In this paper we show that a polynomial-time approximation scheme (PTAS) exists (assuming constant highway dimension). We also show that the considered problems are NP-hard on graphs of highway dimension 1.

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Travelling on Graphs with Small Highway Dimension

Cited by 3 publications

References 49 publications

Near-Linear Time Approximations Schemes for Clustering in Doubling Metrics

Near-Linear Time Approximations Schemes for Clustering in Doubling Metrics

Travelling on Graphs with Small Highway Dimension

Polynomial time approximation schemes for clustering in low highway dimension graphs

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