Minesweeper is NP-complete

Kaye, Robert

doi:10.1007/bf03025367

Cited by 103 publications

(66 citation statements)

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“…A number of other popular one-player computer games have been shown to be NP-hard, most notably Minesweeper-or, more precisely, the Minesweeper "consistency" problem [6]. See the survey of the first author [3] for a summary of other games and puzzles that have been studied from the perspective of computational complexity.…”

Section: Introductionmentioning

confidence: 99%

Tetris is Hard, Even to Approximate

Demaine

Hohenberger

Liben-Nowell

2003

Lecture Notes in Computer Science

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In the popular computer game of Tetris, the player is given a sequence of tetromino pieces and must pack them into a rectangular gameboard initially occupied by a given configuration of filled squares; any completely filled row of the gameboard is cleared and all pieces above it drop by one row. We prove that in the offline version of Tetris, it is NP-complete to maximize the number of cleared rows, maximize the number of tetrises (quadruples of rows simultaneously filled and cleared), minimize the maximum height of an occupied square, or maximize the number of pieces placed before the game ends. We furthermore show the extreme inapproximability of the first and last of these objectives to within a factor of p 1−ε , when given a sequence of p pieces, and the inapproximability of the third objective to within a factor of 2 − ε, for any ε > 0. Our results hold under several variations on the rules of Tetris, including different models of rotation, limitations on player agility, and restricted piece sets.

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Section: Introductionmentioning

confidence: 99%

Tetris is Hard, Even to Approximate

Demaine

Hohenberger

Liben-Nowell

2003

Lecture Notes in Computer Science

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“…Many other popular games and puzzles have been studied from a complexity-theoretic point of view and proved to give rise to hard computational problems, e.g. Tetris [3], Minesweeper [5], Sokoban [2], and Sudoku (also known as Number Place) [6]. We will show that this is also the case for Hiroimono.…”

Section: Examplementioning

confidence: 88%

HIROIMONO is NP-complete

Andersson¹

2007

BRICS

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“…Richard Kaye showed that Minsweeper is NP-complete in 2000 [50]. This result drew quite a lot of attention, as it tangibly connected theoretical computer science to a game everybody had played.…”

Section: Lunar Lockout Lunar Lockoutmentioning

confidence: 99%

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2014

Interpreting Gödel

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There is a fundamental connection between the notions of game and of computation. At its most basic level, this is implied by any game complexity result, but the connection is deeper than this. One example is the concept of alternating nondeterminism, which is intimately connected with two-player games.In the first half of this thesis, I develop the idea of game as computation to a greater degree than has been done previously. I present a general family of games, called Constraint Logic, which is both mathematically simple and ideally suited for reductions to many actual board games. A deterministic version of Constraint Logic corresponds to a novel kind of logic circuit which is monotone and reversible. At the other end of the spectrum, I show that a multiplayer version of Constraint Logic is undecidable. That there are undecidable games using finite physical resources is philosophically important, and raises issues related to the Church-Turing thesis.In the second half of this thesis, I apply the Constraint Logic formalism to many actual games and puzzles, providing new hardness proofs. These applications include sliding-block puzzles, sliding-coin puzzles, plank puzzles, hinged polygon dissections, Amazons, Konane, Cross Purposes, TipOver, and others.

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Minesweeper is NP-complete

Cited by 103 publications

References 0 publications

Tetris is Hard, Even to Approximate

Tetris is Hard, Even to Approximate

HIROIMONO is NP-complete

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