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Approximation Algorithms for Min-Distance Problems

Authors Mina Dalirrooyfard, Virginia Vassilevska Williams, Nikhil Vyas, Nicole Wein, Yinzhan Xu, Yuancheng Yu

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ACM Subject Classification
  • Mathematics of computing → Graph algorithms
Keywords
  • fine-grained complexity
  • graph algorithms
  • diameter
  • radius
  • eccentricities

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Abstract

We study fundamental graph parameters such as the Diameter and Radius in directed graphs, when distances are measured using a somewhat unorthodox but natural measure: the distance between u and v is the minimum of the shortest path distances from u to v and from v to u. The center node in a graph under this measure can for instance represent the optimal location for a hospital to ensure the fastest medical care for everyone, as one can either go to the hospital, or a doctor can be sent to help.
By computing All-Pairs Shortest Paths, all pairwise distances and thus the parameters we study can be computed exactly in O~(mn) time for directed graphs on n vertices, m edges and nonnegative edge weights. Furthermore, this time bound is tight under the Strong Exponential Time Hypothesis [Roditty-Vassilevska W. STOC 2013] so it is natural to study how well these parameters can be approximated in O(mn^{1-epsilon}) time for constant epsilon>0. Abboud, Vassilevska Williams, and Wang [SODA 2016] gave a polynomial factor approximation for Diameter and Radius, as well as a constant factor approximation for both problems in the special case where the graph is a DAG. We greatly improve upon these bounds by providing the first constant factor approximations for Diameter, Radius and the related Eccentricities problem in general graphs. Additionally, we provide a hierarchy of algorithms for Diameter that gives a time/accuracy trade-off.

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Mina Dalirrooyfard, Virginia Vassilevska Williams, Nikhil Vyas, Nicole Wein, Yinzhan Xu, and Yuancheng Yu. Approximation Algorithms for Min-Distance Problems. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 46:1-46:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.46

Author Details

Mina Dalirrooyfard
  • MIT, Cambridge, MA, USA
Virginia Vassilevska Williams
  • MIT, Cambridge, MA, USA
Nikhil Vyas
  • MIT, Cambridge, MA, USA
Nicole Wein
  • MIT, Cambridge, MA, USA
Yinzhan Xu
  • MIT, Cambridge, MA, USA
Yuancheng Yu
  • MIT, Cambridge, MA, USA

Acknowledgements

The authors would like to thank the members of the MIT course 6.S078 open problem sessions, especially Thuy-Duong Vuong, Robin Hui, and Ali Vakilian. These sessions were organized by Erik Demaine, Ryan Williams, and Virginia Vassilevska Williams.

References

  1. Amir Abboud, Fabrizio Grandoni, and Virginia Vassilevska Williams. Subcubic Equivalences Between Graph Centrality Problems, APSP and Diameter. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, CA, USA, January 4-6, 2015, pages 1681-1697, 2015. Google Scholar
  2. Amir Abboud, Virginia Vassilevska Williams, and Joshua R. Wang. Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter in Sparse Graphs. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 377-391, 2016. Google Scholar
  3. D. Aingworth, C. Chekuri, P. Indyk, and R. Motwani. Fast Estimation of Diameter and Shortest Paths (Without Matrix Multiplication). SIAM J. Comput., 28(4):1167-1181, 1999. Google Scholar
  4. Arturs Backurs, Liam Roditty, Gilad Segal, Virginia Vassilevska Williams, and Nicole Wein. Towards tight approximation bounds for graph diameter and eccentricities. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 267-280, 2018. Google Scholar
  5. B. Ben-Moshe, B. K. Bhattacharya, Q. Shi, and A. Tamir. Efficient algorithms for center problems in cactus networks. Theoretical Computer Science, 378(3):237-252, 2007. Google Scholar
  6. P. Berman and S. P. Kasiviswanathan. Faster Approximation of Distances in Graphs. In Proc. WADS, pages 541-552, 2007. Google Scholar
  7. Michele Borassi, Pierluigi Crescenzi, Michel Habib, Walter A. Kosters, Andrea Marino, and Frank W. Takes. Fast diameter and radius BFS-based computation in (weakly connected) real-world graphs: With an application to the six degrees of separation games. Theoretical Computer Science, 2015. accepted. Google Scholar
  8. Massimo Cairo, Roberto Grossi, and Romeo Rizzi. New Bounds for Approximating Extremal Distances in Undirected Graphs. In Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2016, Arlington, VA, USA, January 10-12, 2016, pages 363-376, 2016. Google Scholar
  9. T. M. Chan. All-pairs shortest paths for unweighted undirected graphs in o(mn) time. ACM Transactions on Algorithms, 8(4):34, 2012. Google Scholar
  10. Shiri Chechik, Daniel H. Larkin, Liam Roditty, Grant Schoenebeck, Robert Endre Tarjan, and Virginia Vassilevska Williams. Better Approximation Algorithms for the Graph Diameter. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1041-1052, 2014. Google Scholar
  11. V. Chepoi, F. Dragan, and Y. Vaxès. Center and diameter problems in plane triangulations and quadrangulations. In Proc. SODA, pages 346-355, 2002. Google Scholar
  12. V. Chepoi and F. F. Dragan. A Linear-Time Algorithm for Finding a Central Vertex of a Chordal Graph. In ESA, pages 159-170, 1994. Google Scholar
  13. F. R. K. Chung. Diameters of graphs: Old problems and new results. Congr. Numer., 60:295-317, 1987. Google Scholar
  14. D.G. Corneil, F.F. Dragan, M. Habib, and C. Paul. Diameter determination on restricted graph families. Discr. Appl. Math., 113:143-166, 2001. Google Scholar
  15. L. Cowen and C. Wagner. Compact roundtrip routing for digraphs. In SODA, pages 885-886, 1999. Google Scholar
  16. Mina Dalirrooyfard, Virginia Vassilevska Williams, Nikhil Vyas, Nicole Wein, Yinzhan Xu, and Yuancheng Yu. Approximation Algorithms for Min-Distance Problems, 2019. URL: http://arxiv.org/abs/1904.11606.
  17. D. Dvir and G. Handler. The absolute center of a network. Networks, 43:109-118, 2004. Google Scholar
  18. D. Eppstein. Subgraph isomorphism in planar graphs and related problems. J. Graph Algorithms and Applications, 3(3):1-27, 1999. Google Scholar
  19. Silvio Frischknecht, Stephan Holzer, and Roger Wattenhofer. Networks cannot compute their diameter in sublinear time. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 1150-1162. SIAM, 2012. Google Scholar
  20. S.L. Hakimi. Optimum location of switching centers and absolute centers and medians of a graph. Oper. Res., 12:450-459, 1964. Google Scholar
  21. Seth Pettie. A faster all-pairs shortest path algorithm for real-weighted sparse graphs. In International Colloquium on Automata, Languages, and Programming, pages 85-97. Springer, 2002. Google Scholar
  22. Seth Pettie and Vijaya Ramachandran. A Shortest Path Algorithm for Real-Weighted Undirected Graphs. SIAM J. Comput., 34(6):1398-1431, 2005. URL: http://dx.doi.org/10.1137/S0097539702419650.
  23. Liam Roditty and Virginia Vassilevska Williams. Fast approximation algorithms for the diameter and radius of sparse graphs. In Proceedings of the 45th annual ACM symposium on Symposium on theory of computing, STOC '13, pages 515-524, New York, NY, USA, 2013. ACM. URL: http://dx.doi.org/10.1145/2488608.2488673.
  24. O. Weimann and R. Yuster. Approximating the Diameter of Planar Graphs in Near Linear Time. In Proc. ICALP, 2013. Google Scholar
  25. Ryan Williams. Faster all-pairs shortest paths via circuit complexity. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages 664-673, 2014. Google Scholar
  26. C. Wulff-Nilsen. Wiener index, diameter, and stretch factor of a weighted planar graph in subquadratic time. Technical report, University of Copenhagen, 2008. Google Scholar
  27. Raphael Yuster. Computing the diameter polynomially faster than APSP. arXiv preprint, 2010. URL: http://arxiv.org/abs/1011.6181.
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