The Computational Complexity of RaceTrack

Martin Gardner in the early 1970's described the game of RaceTrack [M. Gardner, Mathematical games--Sim, Chomp and Race Track: new games for the intellect (and not for Lady Luck), Scientific American, 228(1):108-115, Jan. 1973]. Here we study the complexity of deciding whether a RACETRACK player has a winning strategy. We first prove that the complexity of RACETRACK reachability, i.e., whether the finish line can be reached or not, crucially depends on whether the car can touch the edge of the carriageway (racetrack): the non-touching variant is NL-complete while the touching variant is equivalent to the undirected grid graph reachability problem, a problem in L but not known to be L-hard. Then we show that single-player RACETRACK is NL-complete, regardless of whether driving on the track boundary is allowed or not, and that deciding the existence of a winning strategy in Gardner's original two-player game is P-complete. Hence RACETRACK is an example of a game that is interesting to play despite the fact that deciding the existence of a winning strategy is most likely not NP-hard.