FUN 2012: Fun with Algorithms pp 210-222

# On the Complexity of Rolling Block and Alice Mazes

• Markus Holzer
• Sebastian Jakobi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7288)

## Abstract

We investigate the computational complexity of two maze problems, namely rolling block and Alice mazes. Simply speaking, in the former game one has to roll blocks through a maze, ending in a particular game situation, and in the latter one, one has to move tokens of variable speed through a maze following some prescribed directions. It turns out that when the number of blocks or the number of tokens is not restricted (unbounded), then the problem of solving such a maze becomes PSPACE-complete. Hardness is shown via a reduction from the nondeterministic constraint logic (NCL) of [E. D. Demaine, R. A. Hearn: A uniform framework or modeling computations as games. Proc. CCC, 2008] to the problems in question. By using only blocks of size 2×1×1, and no forbidden squares, we improve a previous result of [K. Buchin, M. Buchin: Rolling block mazes are PSPACE-complete. J. Inform. Proc., 2012] on rolling block mazes to best possible. Moreover, we also consider bounded variants of these maze games, i.e., when the number of blocks or tokens is bounded by a constant, and prove close relations to variants of graph reachability problems.

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