Fine-grained Lower Bounds on Cops and Robbers
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.
Cops and Robbers
Mathematics of computing~Graph theory
9:1-9:12
Regular Paper
Sebastian
Brandt
Sebastian Brandt
ETH Zürich, Zürich, Switzerland
Seth
Pettie
Seth Pettie
University of Michigan, Ann Arbor, MI, USA
Supported by NSF grants CCF-1514383 and CCF-1637546.
Jara
Uitto
Jara Uitto
ETH Zürich, Zürich, Switzerland
Partially supported by ERC Grant No. 336495 (ACDC).
10.4230/LIPIcs.ESA.2018.9
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Sebastian Brandt, Seth Pettie, and Jara Uitto
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