A Timecop’s Work Is Harder Than You Think

Authors Nils Morawietz, Carolin Rehs, Mathias Weller

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Nils Morawietz
  • Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Germany
Carolin Rehs
  • Institute of Computer Science, Heinrich Heine Universität, Düsseldorf, Germany
Mathias Weller
  • CNRS, LIGM, Université Gustave Eiffel, Marne-la-Vallée, France


We thank our anonymous reviewers for their various helpful suggestions, in particular the connection to Tally-DFAs.

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Nils Morawietz, Carolin Rehs, and Mathias Weller. A Timecop’s Work Is Harder Than You Think. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 71:1-71:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We consider the (parameterized) complexity of a cop and robber game on periodic, temporal graphs and a problem on periodic sequences to which these games relate intimately. In particular, we show that it is NP-hard to decide (a) whether there is some common index at which all given periodic, binary sequences are 0, and (b) whether a single cop can catch a single robber on an edge-periodic temporal graph. We further present results for various parameterizations of both problems and show that hardness not only applies in general, but also for highly limited instances. As one main result we show that even if the graph has a size-2 vertex cover and is acyclic in each time step, the cop and robber game on periodic, temporal graphs is NP-hard and W[1]-hard when parameterized by the size of the underlying input graph.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Dynamic graph algorithms
  • edge-periodic temporal graphs
  • cops and robbers
  • tally-intersection
  • congruence satisfyability


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