Paper by Erik D. Demaine

Reference:
Erik D. Demaine, Fermi Ma, Erik Waingarten, Ariel Schvartzman, and Scott Aaronson, “The Fewest Clues Problem”, in Proceedings of the 8th International Conference on Fun with Algorithms (FUN 2016), La Maddalena, Italy, June 8–10, 2016, 12:1–12:12.

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 Σ2P-complete. Along the way, we show that the FCP versions of 3SAT, 1-in-3 SAT, Triangle Partition, Planar 3SAT, and Latin Square are all Σ2P-complete. We show that even problems in P have difficult FCP versions, sometimes even Σ2P-complete, though “closed under cluing” problems are in the (presumably) smaller class NP; for example, FCP 2SAT is NP-complete.

Availability:
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Last updated September 3, 2017 by Erik Demaine.