Abstract

Graphs with bounded highway dimension were introduced by Abraham et al. [Proceedings of SODA 2010, pp. 782--793] as a model of transportation networks. We show that any such graph can be embedded into a distribution over bounded treewidth graphs with arbitrarily small distortion. More concretely, given a weighted graph $G=(V,E)$ of constant highway dimension, we show how to randomly compute a weighted graph $H=(V,E')$ that distorts shortest path distances of $G$ by at most a $1+\varepsilon$ factor in expectation, and whose treewidth is polylogarithmic in the aspect ratio of $G$. Our probabilistic embedding implies quasi-polynomial time approximation schemes for a number of optimization problems that naturally arise in transportation networks, including Travelling Salesman, Steiner Tree, and Facility Location. To construct our embedding for low highway dimension graphs we extend Talwar's [Proceedings of STOC 2004, pp. 281--290] embedding of low doubling dimension metrics into bounded treewidth graphs, which generalizes known results for Euclidean metrics. We add several nontrivial ingredients to Talwar's techniques, and in particular thoroughly analyze the structure of low highway dimension graphs. Thus we demonstrate that the geometric toolkit used for Euclidean metrics extends beyond the class of low doubling metrics.

Keywords

  1. metric embeddings
  2. highway dimension
  3. QPTAS
  4. travelling salesman
  5. Steiner tree
  6. facility location

MSC codes

  1. 68W25
  2. 68Q25

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References

1.
I. Abraham, A. Fiat, A. V. Goldberg, and R. F. Werneck, Highway dimension, shortest paths, and provably efficient algorithms, in Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms, 2010, pp. 782--793, https://doi.org/10.1137/1.9781611973075.64.
2.
I. Abraham, D. Delling, A. Fiat, A. V. Goldberg, and R. F. Werneck, VC-dimension and shortest path algorithms, in Automata, Languages and Programming, Springer, New York, 2011, pp. 690--699, https://doi.org/10.1007/978-3-642-22006-7_58.
3.
I. Abraham, C. Gavoille, A. Gupta, O. Neiman, and K. Talwar, Cops, robbers, and threatening skeletons: Padded decomposition for minor-free graphs, in Proceedings of the 46th Annual ACM Symposium on Theory of Computing, 2014, pp. 79--88, https://doi.org/10.1145/2591796.2591849.
4.
I. Abraham, D. Delling, A. Fiat, A. V. Goldberg, and R. F. Werneck, Highway dimension and provably efficient shortest path algorithms, J. ACM, 63 (2016), 41.
5.
A. A. Ageev, An approximation scheme for the uncapacitated facility location problem on planar graphs, Citeseer.
6.
A. A. Ageev, A criterion of polynomial-time solvability for the network location problem, in Integer Programming and Combinatorial Optimization, Carnegie Mellon University, Pittsburgh, PA, 1992, pp. 237--245.
7.
S. Arora, Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems, J. ACM, 45 (1998), pp. 753--782, https://doi.org/10.1145/290179.290180.
8.
S. Arora, Approximation schemes for NP-hard geometric optimization problems: A survey, Math. Program., 97 (2003), pp. 43--69, https://doi.org/10.1007/s10107-003-0438-y.
9.
S. Arora, P. Raghavan, and S. Rao, Approximation schemes for Euclidean k-medians and related problems, in Proceedings of the 30th Annual ACM Symposium on Theory of Computing, ACM, New York, 1998, pp. 106--113, https://doi.org/10.1145/276698.276718.
10.
S. Arora, Polynomial time approximation schemes for Euclidean TSP and other geometric problems, in Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, 1996, pp. 2--11, https://doi.org/10.1109/SFCS.1996.548458.
11.
Y. Bartal, Probabilistic approximation of metric spaces and its algorithmic applications, in Proceedings of the 37th Annual Symposium on Foundations of Computer Science, IEEE, 1996, pp. 184--193, https://doi.org/10.1109/SFCS.1996.548477.
12.
Y. Bartal, On approximating arbitrary metrices by tree metrics, in Proceedings of the 30th Annual ACM Symposium on Theory of Computing, ACM, New York, 1998, pp. 161--168, https://doi.org/10.1145/276698.276725.
13.
Y. Bartal, L.-A. Gottlieb, and R. Krauthgamer, The traveling salesman problem: Low-dimensionality implies a polynomial time approximation scheme, in Proceedings of the 44th Annual ACM Symposium on Theory of Computing, ACM, New York, 2012, pp. 663--672, https://doi.org/10.1145/2213977.2214038.
14.
H. Bast, S. Funke, D. Matijevic, P. Sanders, and D. Schultes, In transit to constant time shortest-path queries in road networks, in Proceedings of the Algorithm Engineering and Experiments, SIAM, Philadelphia, 2007, https://doi.org/10.1137/1.9781611972870.5.
15.
H. Bast, S. Funke, and D. Matijevic, Ultrafast shortest-path queries via transit nodes, in The Shortest Path Problem: 9th DIMACS Implementation Challenge, DIMACS Ser. Discrete Math. Theoret. Sci. 74, 2009, pp. 175--192.
16.
M. Bateni, C. Chekuri, A. Ene, M. T. Hajiaghayi, N. Korula, and D. Marx, Prize-collecting Steiner problems on planar graphs, in Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2011, pp. 1028--1049, https://doi.org/10.1137/1.9781611973082.79.
17.
R. Bauer, T. Columbus, I. Rutter, and D. Wagner, Search-space size in contraction hierarchies, in Automata, Languages, and Programming, Springer, New York, 2013, pp. 93--104, https://doi.org/10.1007/978-3-642-39206-1_9.
18.
G. Borradaile, C. Kenyon-Mathieu, and P. Klein, A polynomial-time approximation scheme for Steiner tree in planar graphs, in Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, 2007, pp. 1285--1294.
19.
M. Brazil, R. L. Graham, D. A. Thomas, and M. Zachariasen, On the history of the Euclidean Steiner tree problem, Arch. Hist. Exact Sci., 68 (2014), pp. 327--354, https://doi.org/10.1007/s00407-013-0127-z.
20.
D. E. Carroll and A. Goel, Lower bounds for embedding into distributions over excluded minor graph families, in Proceedings of the 12th European Symposium on Algorithms, Springer, New York, 2004, pp. 146--156, https://doi.org/10.1007/978-3-540-30140-0_15.
21.
A. Chakrabarti, A. Jaffe, J. Lee, and J. Vincent, Local moves and lossy invariants in planar graph embeddings, in Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science, 2008.
22.
T.-H. Chan and A. Gupta, Approximating TSP on metrics with bounded global growth, SIAM J. Comput., 41 (2012), pp. 587--617.
23.
M. Chlebík and J. Chlebíková, Approximation hardness of the Steiner tree problem on graphs, in Scandinavian Workshop an Algorithm Theory, Springer, New York, 2002, pp. 170--179, https://doi.org/10.1007/3-540-45471-3_18.
24.
W. Cook, In Pursuit of the Traveling Salesman: Mathematics at the Limits of Computation, Princeton University Press, Princeton, NJ, 2012.
25.
L. Engebretsen and M. Karpinski, Approximation hardness of TSP with bounded metrics, in Proceedings of the 28th International Colloquium on Automata, Languages and Programming, Springer, New York, 2001, pp. 201--212, https://doi.org/10.1007/3-540-48224-5_17.
26.
J. Fakcharoenphol, S. Rao, and K. Talwar, A tight bound on approximating arbitrary metrics by tree metrics, in Proceedings of the 35th Annual ACM Symposium on Theory of Computing, ACM, New York, 2003, pp. 448--455.
27.
A. E. Feldmann, Fixed parameter approximations for k-center problems in low highway dimension graphs, in Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming, 2015, pp. 588--600, https://doi.org/10.1007/978-3-662-47666-6_47.
28.
M. R. Garey and D. S. Johnson, Computers and Intractability, W. H. Freeman, San Francisco, 2002.
29.
S. Guha and S. Khuller, Greedy strikes back: Improved facility location algorithms, J. Algorithms, 31 (1999), pp. 228--248, https://doi.org/10.1006/jagm.1998.0993.
30.
A. Gupta, R. Krauthgamer, and J. R. Lee, Bounded geometries, fractals, and low-distortion embeddings, in Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, IEEE, 2003, pp. 534--543, https://doi.org/10.1109/SFCS.2003.1238226.
31.
P. Klein, A linear-time approximation scheme for TSP in undirected planar graphs with edge-weights, SIAM J. Comput., 37 (2008), pp. 1926--1952, https://doi.org/10.1137/060649562.
32.
A. Kosowski and L. Viennot, Beyond highway dimension: Small distance labels using tree skeletons, in Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2017, pp. 1462--1478.
33.
M. Mahdian, Y. Ye, and J. Zhang, Approximation algorithms for metric facility location problems, SIAM J. Comput., 36 (2006), pp. 411--432, https://doi.org/10.1137/S0097539703435716.
34.
J. S. B. Mitchell, Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems, SIAM J. Comput., 28 (1999), pp. 1298--1309, https://doi.org/10.1137/S0097539796309764.
35.
N. Robertson and P. D. Seymour, Graph minors. II. Algorithmic aspects of tree-width, J. Algorithms, 7 (1986), pp. 309--322, https://doi.org/10.1016/0196-6774(86)90023-4.
36.
A. Schrijver, On the history of combinatorial optimization (till $1960$), in Handbooks in Operations Research and Management Science, Discrete Optimization 12, 2005, pp. 1--68.
37.
H. K. Smith, G. Laporte, and P. R. Harper, Locational analysis: Highlights of growth to maturity, J. Oper. Res. Soc., 60 (2009), pp. S140--S148, https://doi.org/10.1057/jors.2008.172.
38.
K. Talwar, Bypassing the embedding: Algorithms for low dimensional metrics, in Proceedings of the 36th Annual ACM Symposium on Theory of Computing, ACM, New York, 2004, pp. 281--290, https://doi.org/10.1145/1007352.1007399.
39.
V. V. Vazirani, Approximation Algorithms, Springer, New York, 2001.

Information & Authors

Information

Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 1667 - 1704
ISSN (online): 1095-7111

History

Submitted: 23 March 2016
Accepted: 12 June 2018
Published online: 21 August 2018

Keywords

  1. metric embeddings
  2. highway dimension
  3. QPTAS
  4. travelling salesman
  5. Steiner tree
  6. facility location

MSC codes

  1. 68W25
  2. 68Q25

Authors

Affiliations

Funding Information

Czech Science Foundation : P202/12/G061
Hausdorff Research Institute for Mathematics
Research Institute for Discrete Mathematics in Bonn, Germany
Natural Sciences and Engineering Research Council of Canada https://doi.org/10.13039/501100000038

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