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Algorithms 2012, 5(1), 148-157; doi:10.3390/a5010148

Article

Any Monotone Function Is Realized by Interlocked Polygons

Erik D. Demaine ¹ , Martin L. Demaine ¹  and Ryuhei Uehara ^2,*

¹ Computer Science and Artificial Intelligence Lab, Massachusetts Institute of Technology, MA 02139,USA ² School of Information Science, Japan Advanced Institute of Science and Technology, Ishikawa 923-1292, Japan

* Author to whom correspondence should be addressed.

Received: 15 November 2011; in revised form: 13 March 2012 / Accepted: 14 March 2012 / Published: 19 March 2012

(This article belongs to the Special Issue Puzzle/Game Algorithms)

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Abstract: Suppose there is a collection of n simple polygons in the plane, none of which overlap each other. The polygons are interlocked if no subset can be separated arbitrarily far from the rest. It is natural to ask the characterization of the subsets that makes the set of interlocked polygons free (not interlocked). This abstracts the essence of a kind of sliding block puzzle. We show that any monotone Boolean function ƒ on n variables can be described by m = O(n) interlocked polygons. We also show that the decision problem that asks if given polygons are interlocked is PSPACE-complete.

Keywords: computational complexity; interlocked polygons; monotone boolean function; sliding block puzzle

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