Tight Approximation Algorithms for Bichromatic Graph Diameter and Related Problems
Some of the most fundamental and well-studied graph parameters are the Diameter (the largest shortest paths distance) and Radius (the smallest distance for which a "center" node can reach all other nodes). The natural and important ST-variant considers two subsets S and T of the vertex set and lets the ST-diameter be the maximum distance between a node in S and a node in T, and the ST-radius be the minimum distance for a node of S to reach all nodes of T. The bichromatic variant is the special case in which S and T partition the vertex set.
In this paper we present a comprehensive study of the approximability of ST and Bichromatic Diameter, Radius, and Eccentricities, and variants, in graphs with and without directions and weights. We give the first nontrivial approximation algorithms for most of these problems, including time/accuracy trade-off upper and lower bounds. We show that nearly all of our obtained bounds are tight under the Strong Exponential Time Hypothesis (SETH), or the related Hitting Set Hypothesis.
For instance, for Bichromatic Diameter in undirected weighted graphs with m edges, we present an O~(m^{3/2}) time 5/3-approximation algorithm, and show that under SETH, neither the running time, nor the approximation factor can be significantly improved while keeping the other unchanged.
approximation algorithms
fine-grained complexity
diameter
radius
eccentricities
Theory of computation~Graph algorithms analysis
Theory of computation~Approximation algorithms analysis
Theory of computation~Problems, reductions and completeness
Theory of computation~Shortest paths
47:1-47:15
Track A: Algorithms, Complexity and Games
https://arxiv.org/abs/1904.11601
The authors would like to thank Arturs Backurs for discussions during the early stages of this work.
Mina
Dalirrooyfard
Mina Dalirrooyfard
MIT, Cambridge, MA, USA
Supported by an Akamai Presidential Fellowship and NSF Grant CCF-6936484.
Virginia Vassilevska
Williams
Virginia Vassilevska Williams
MIT, Cambridge, MA, USA
Supported by an NSF CAREER Award, NSF Grants CCF-1417238, CCF-1528078 and CCF-1514339, a BSF Grant BSF:2012338 and a Sloan Research Fellowship.
Nikhil
Vyas
Nikhil Vyas
MIT, Cambridge, MA, USA
Supported by an Akamai Presidential Fellowship and NSF Grant CCF-1552651.
Nicole
Wein
Nicole Wein
MIT, Cambridge, MA, USA
Supported by an NSF Graduate Fellowship and NSF Grant CCF-1514339.
10.4230/LIPIcs.ICALP.2019.47
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.
Pankaj K. Agarwal, Herbert Edelsbrunner, Otfried Schwarzkopf, and Emo Welzl. Euclidean Minimum Spanning Trees and Bichromatic Closest Pairs. Discrete Comput. Geom., 6(1):407-422, December 1991.
D. Aingworth, C. Chekuri, P. Indyk, and R. Motwani. Fast Estimation of Diameter and Shortest Paths (Without Matrix Multiplication). SIAM Journal on Computing, 28(4):1167-1181, 1999.
Arturs Backurs, Liam Roditty, Gilad Segal, Virginia Vassilevska Williams, and Nicole Wein. Towards Tight Approximation Bounds for Graph Diameter and Eccentricities. In Proceedings of STOC'18, page to appear, 2018.
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.
T. M. Chan. More Algorithms for All-Pairs Shortest Paths in Weighted Graphs. In Proc. STOC, pages 590-598, 2007.
Bernard Chazelle, Herbert Edelsbrunner, Leonidas J Guibas, and Micha Sharir. Algorithms for bichromatic line-segment problems and polyhedral terrains. Algorithmica, 11(2):116-132, 1994.
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.
Don Coppersmith and Michael Elkin. Sparse sourcewise and pairwise distance preservers. SIAM Journal on Discrete Mathematics, 20(2):463-501, 2006.
Pierluigi Crescenzi, Roberto Grossi, Leonardo Lanzi, and Andrea Marino. On Computing the Diameter of Real-World Directed (Weighted) Graphs. In Ralf Klasing, editor, Experimental Algorithms: 11th International Symposium, SEA 2012, Bordeaux, France, June 7-9, 2012. Proceedings, pages 99-110, Berlin, Heidelberg, 2012. Springer Berlin Heidelberg.
Marek Cygan, Fabrizio Grandoni, and Telikepalli Kavitha. On pairwise spanners. arXiv preprint, 2013. URL: http://arxiv.org/abs/1301.1999.
http://arxiv.org/abs/1301.1999
Mina Dalirrooyfard, Virginia Vassilevska Williams, Nikhil Vyas, and Nicole Wein. Tight Approximation Algorithms for Bichromatic Graph Diameter and Related Problems, 2019. URL: http://arxiv.org/abs/1904.11601.
http://arxiv.org/abs/1904.11601
Adrian Dumitrescu and Sumanta Guha. Extreme Distances in Multicolored Point Sets. J. Graph Algorithms and Applications, 8(1):27-38, 2004.
Jiawei Gao, Russell Impagliazzo, Antonina Kolokolova, and R. Ryan Williams. Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, Barcelona, Spain, Hotel Porta Fira, January 16-19, pages 2162-2181, 2017.
R. Impagliazzo and R. Paturi. On the Complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001.
Piotr Indyk. A near linear time constant factor approximation for Euclidean bichromatic matching (cost). In SODA, volume 7, pages 39-42, 2007.
Naoki Katoh and Kazuo Iwano. Finding K Farthest Pairs and K Closest/Farthest Bichromatic Pairs for Points in the Plane. In Proceedings of the Eighth Annual Symposium on Computational Geometry, SCG '92, pages 320-329, 1992.
Telikepalli Kavitha. New pairwise spanners. Theory of Computing Systems, 61(4):1011-1036, 2017.
Philip N Klein. A subset spanner for planar graphs, with application to subset TSP. In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 749-756. ACM, 2006.
ClĂ©mence Magnien, Matthieu Latapy, and Michel Habib. Fast Computation of Empirically Tight Bounds for the Diameter of Massive Graphs. J. Exp. Algorithmics, 13:10:1.10-10:1.9, February 2009.
David Peleg, Liam Roditty, and Elad Tal. Distributed Algorithms for Network Diameter and Girth. In Automata, Languages, and Programming: 39th International Colloquium, ICALP 2012, Warwick, UK, July 9-13, 2012, Proceedings, Part II, pages 660-672, 2012.
S. Pettie. A new approach to all-pairs shortest paths on real-weighted graphs. Theor. Comput. Sci., 312(1):47-74, 2004.
Seth Pettie and Vijaya Ramachandran. A Shortest Path Algorithm for Real-Weighted Undirected Graphs. SIAM J. Comput., 34(6):1398-1431, 2005.
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.
http://dx.doi.org/10.1145/2488608.2488673
Frank W. Takes and Walter A. Kosters. Determining the Diameter of Small World Networks. In Proceedings of the 20th ACM International Conference on Information and Knowledge Management, CIKM '11, pages 1191-1196, 2011.
Virginia Vassilevska Williams. Hardness of Easy Problems: Basing Hardness on Popular Conjectures such as the Strong Exponential Time Hypothesis (Invited Talk). In 10th International Symposium on Parameterized and Exact Computation, IPEC 2015, September 16-18, 2015, Patras, Greece, pages 17-29, 2015.
R. Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theor. Comput. Sci., 348(2-3):357-365, 2005.
Ryan Williams. On the Difference Between Closest, Furthest, and Orthogonal Pairs: Nearly-linear vs Barely-subquadratic Complexity. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '18, pages 1207-1215, 2018.
A. Yao. On Constructing Minimum Spanning Trees in k-Dimensional Spaces and Related Problems. SIAM Journal on Computing, 11(4):721-736, 1982.
Mina Dalirrooyfard, Virginia Vassilevska Williams, Nikhil Vyas, and Nicole Wein
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode