Dynamic Complexity of Reachability: How Many Changes Can We Handle?

Authors Samir Datta, Pankaj Kumar, Anish Mukherjee , Anuj Tawari, Nils Vortmeier, Thomas Zeume

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Author Details

Samir Datta
  • Chennai Mathematical Institute, India
Pankaj Kumar
  • Chennai Mathematical Institute, India
  • Department of Applied Mathematics, Charles University, Prague, Czech Republic
Anish Mukherjee
  • Institute of Informatics, University of Warsaw, Poland
Anuj Tawari
  • Chennai Mathematical Institute, India
Nils Vortmeier
  • TU Dortmund, Germany
Thomas Zeume
  • Ruhr-Universität Bochum, Germany


The first author would like to thank Chetan Gupta for finding a problem in a previous version of the proof of Theorem 14.

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Samir Datta, Pankaj Kumar, Anish Mukherjee, Anuj Tawari, Nils Vortmeier, and Thomas Zeume. Dynamic Complexity of Reachability: How Many Changes Can We Handle?. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 122:1-122:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


In 2015, it was shown that reachability for arbitrary directed graphs can be updated by first-order formulas after inserting or deleting single edges. Later, in 2018, this was extended for changes of size (log n)/(log log n), where n is the size of the graph. Changes of polylogarithmic size can be handled when also majority quantifiers may be used. In this paper we extend these results by showing that, for changes of polylogarithmic size, first-order update formulas suffice for maintaining (1) undirected reachability, and (2) directed reachability under insertions. For classes of directed graphs for which efficient parallel algorithms can compute non-zero circulation weights, reachability can be maintained with update formulas that may use "modulo 2" quantifiers under changes of polylogarithmic size. Examples for these classes include the class of planar graphs and graphs with bounded treewidth. The latter is shown here. As the logics we consider cannot maintain reachability under changes of larger sizes, our results are optimal with respect to the size of the changes.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • Dynamic complexity
  • reachability
  • complex changes


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