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Fine-grained Lower Bounds on Cops and Robbers

Authors Sebastian Brandt, Seth Pettie, Jara Uitto

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ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • Cops and Robbers

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Abstract

Cops and Robbers is a classic pursuit-evasion game played between a group of g cops and one robber on an undirected N-vertex graph G. We prove that the complexity of deciding the winner in the game under optimal play requires Omega (N^{g-o(1)}) time on instances with O(N log^2 N) edges, conditioned on the Strong Exponential Time Hypothesis. Moreover, the problem of calculating the minimum number of cops needed to win the game is 2^{Omega (sqrt{N})}, conditioned on the weaker Exponential Time Hypothesis. Our conditional lower bound comes very close to a conditional upper bound: if Meyniel's conjecture holds then the cop number can be decided in 2^{O(sqrt{N}log N)} time.
In recent years, the Strong Exponential Time Hypothesis has been used to obtain many lower bounds on classic combinatorial problems, such as graph diameter, LCS, EDIT-DISTANCE, and REGEXP matching. To our knowledge, these are the first conditional (S)ETH-hard lower bounds on a strategic game.

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Sebastian Brandt, Seth Pettie, and Jara Uitto. Fine-grained Lower Bounds on Cops and Robbers. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 9:1-9:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.ESA.2018.9

Author Details

Sebastian Brandt
  • ETH Zürich, Zürich, Switzerland
Seth Pettie
  • University of Michigan, Ann Arbor, MI, USA
Jara Uitto
  • ETH Zürich, Zürich, Switzerland

Funding

  • Pettie, Seth: Supported by NSF grants CCF-1514383 and CCF-1637546.
  • Uitto, Jara: Partially supported by ERC Grant No. 336495 (ACDC).

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