Algorithms and Hardness for Diameter in Dynamic Graphs
The diameter, radius and eccentricities are natural graph parameters. While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each update or from solving dynamic All-Pairs Shortest Paths (APSP), which is very computationally intensive. This is the situation for dynamic approximation algorithms as well, and even if only edge insertions or edge deletions need to be supported.
This paper provides a comprehensive study of the dynamic approximation of Diameter, Radius and Eccentricities, providing both conditional lower bounds, and new algorithms whose bounds are optimal under popular hypotheses in fine-grained complexity. Some of the highlights include:
- Under popular hardness hypotheses, there can be no significantly better fully dynamic approximation algorithms than recomputing the answer after each update, or maintaining full APSP.
- Nearly optimal partially dynamic (incremental/decremental) algorithms can be achieved via efficient reductions to (incremental/decremental) maintenance of Single-Source Shortest Paths. For instance, a nearly (3/2+epsilon)-approximation to Diameter in directed or undirected n-vertex, m-edge graphs can be maintained decrementally in total time m^{1+o(1)}sqrt{n}/epsilon^2. This nearly matches the static 3/2-approximation algorithm for the problem that is known to be conditionally optimal.
fine-grained complexity
graph algorithms
dynamic algorithms
Mathematics of computing~Graph algorithms
13:1-13:14
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1811.12527.
The authors would like to thank Roei Tov for discussions.
Bertie
Ancona
Bertie Ancona
MIT, Cambridge, MA, USA
Monika
Henzinger
Monika Henzinger
University of Vienna, Austria
The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement No. 340506.
Liam
Roditty
Liam Roditty
Bar Ilan University, Ramat Gan, Israel
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.
Nicole
Wein
Nicole Wein
MIT, Cambridge, MA, USA
Supported by an NSF Graduate Fellowship and NSF Grant CCF-1514339.
10.4230/LIPIcs.ICALP.2019.13
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Bertie Ancona, Monika Henzinger, Liam Roditty, Virginia Vassilevska Williams, and Nicole Wein
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