License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.FUN.2016.12
URN: urn:nbn:de:0030-drops-58654
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2016/5865/
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Demaine, Erik D. ; Ma, Fermi ; Schvartzman, Ariel ; Waingarten, Erik ; Aaronson, Scott

The Fewest Clues Problem

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Abstract

When analyzing the computational complexity of well-known puzzles, most papers consider the algorithmic challenge of solving a given instance of (a generalized form of) the puzzle. We take a different approach by analyzing the computational complexity of designing a "good" puzzle. We assume a puzzle maker designs part of an instance, but before publishing it, wants to ensure that the puzzle has a unique solution. Given a puzzle, we introduce the FCP (fewest clues problem) version of the problem:

Given an instance to a puzzle, what is the minimum number of clues we must add in order to make the instance uniquely solvable?

We analyze this question for the Nikoli puzzles Sudoku, Shakashaka, and Akari. Solving these puzzles is NP-complete, and we show their FCP versions are Sigma_2^P-complete. Along the way, we show that the FCP versions of 3SAT, 1-in-3SAT, Triangle Partition, Planar 3SAT, and Latin Square are all Sigma_2^P-complete. We show that even problems in P have difficult FCP versions, sometimes even Sigma_2^P-complete, though "closed under cluing" problems are in the (presumably) smaller class NP; for example, FCP 2SAT is NP-complete.

BibTeX - Entry

@InProceedings{demaine_et_al:LIPIcs:2016:5865,
  author =	{Erik D. Demaine and Fermi Ma and Ariel Schvartzman and Erik Waingarten and Scott Aaronson},
  title =	{{The Fewest Clues Problem}},
  booktitle =	{8th International Conference on Fun with Algorithms (FUN 2016)},
  pages =	{12:1--12:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-005-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{49},
  editor =	{Erik D. Demaine and Fabrizio Grandoni},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/5865},
  URN =		{urn:nbn:de:0030-drops-58654},
  doi =		{10.4230/LIPIcs.FUN.2016.12},
  annote =	{Keywords: computational complexity, pencil-and-paper puzzles, hardness reductions}
}

Keywords: computational complexity, pencil-and-paper puzzles, hardness reductions
Collection: 8th International Conference on Fun with Algorithms (FUN 2016)
Issue Date: 2016
Date of publication: 02.06.2016

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