Paper by Erik D. Demaine

Reference:

Erik D. Demaine, Fermi Ma, Erik Waingarten, Ariel Schvartzman, and Scott Aaronson, “The Fewest Clues Problem”, Theoretical Computer Science, volume 748, November 2018, pages 28–39.

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 Σ₂^P-complete. Along the way, we show that the FCP versions of Triangle Partition, Planar 1-in-3 SAT, and Latin Square are all Σ₂^P-complete. We show that even problems in P have difficult FCP versions, sometimes even Σ₂^P-complete, though “closed under cluing” problems are in the (presumably) smaller class NP; for example, FCP 2SAT is NP-complete.

Comments:

This paper is also available from ScienceDirect.

Availability:

The paper is available in PDF (560k).

See information on file formats.

[Google Scholar search]

Related papers:

FewestClues_FUN2016 (The Fewest Clues Problem)