super mario bros. is np-hardcourses.csail.mit.edu/6.892/spring19/lectures/l06_images.pdf ·...

Super Mario Bros. is NP-Hard[Aloupis, Demaine, Guo, Viglietta 2014]

𝑥𝑥 OR ¬𝑦𝑦 OR 𝑧𝑧 & 𝑥𝑥 OR 𝑦𝑦 OR ¬𝑦𝑦 &(¬𝑥𝑥 OR ¬𝑦𝑦 OR ¬𝑧𝑧) & (¬𝑥𝑥 OR ¬𝑦𝑦 OR ¬𝑧𝑧)

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http://arxiv.org/abs/1203.1895

[Bern & Hayes 1996]

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Figure 13.7 of GFALOP

Akari / Light Up [Nikoli 2001]

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http://www.nikoli.com/en/puzzles/bijutsukan/ Play more here: http://www.puzzle-light-up.com/

Akari / Light Up is NP-complete[McPhail 2005]

wire gadget

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https://web.archive.org/web/20141218232731/http://people.cs.umass.edu/~mcphailb/papers/2005lightup.pdf


split/negationgadget

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split gadget NOT gate

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OR/XNOR gate

crossover gadget

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¬𝑥𝑥 ∨ 𝑥𝑥 ∧ 𝑦𝑦 ∨ 𝑧𝑧

𝑥𝑥

𝑥𝑥 𝑥𝑥

𝑥𝑥𝑥𝑥′

𝑥𝑥′

𝑧𝑧

𝑧𝑧

𝑥𝑥′𝑥𝑥

𝑦𝑦′

𝑦𝑦

𝑦𝑦

∨

∧

∨

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http://courses.csail.mit.edu/6.890/fall14/

𝑥𝑥

𝑥𝑥

Minesweeper Consistency isNP-complete [Kaye 2000]

wire

wire terminator

split,NOT,turn

𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥

𝑥𝑥𝑥𝑥 𝑥𝑥 𝑥𝑥

𝑥𝑥𝑥𝑥

𝑥𝑥𝑥𝑥

𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥

𝑥𝑥 𝑥𝑥

𝑥𝑥𝑥𝑥𝑥𝑥

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Figure drawn by course staff based on http://simon.bailey.at/random/kaye.minesweeper.pdf


NOT

Phase changer(2 NOTs)

𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥

𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥

𝑥𝑥 𝑥𝑥

𝑥𝑥 𝑥𝑥

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AND

turn

𝑥𝑥𝑥𝑥𝑥𝑥

𝑥𝑥

𝑥𝑥𝑥𝑥

𝑥𝑥

𝑧𝑧

𝑥𝑥𝑥𝑥

𝑥𝑥

𝑥𝑥

𝑧𝑧𝑧𝑧

𝑧𝑧

𝑧𝑧𝑧𝑧 𝑧𝑧

𝑥𝑥 𝑥𝑥𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡𝑡𝑡 𝑡𝑡

𝑡𝑡

𝑡𝑡

𝑡𝑡

𝑡𝑡

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crossover

XOR

[Goldschlager 1977]

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Figures 11 and 12 of http://simon.bailey.at/random/kaye.minesweeper.pdf which in turn is based on http://dx.doi.org/10.1145/1008354.1008356

Minesweeper is CoNP-Complete[Scott, Stege, van Rooij 2011]

turn gadgetwire gadget terminator gadget

𝑥𝑥 𝑥𝑥

𝑥𝑥

𝑥𝑥

𝑥𝑥 𝑥𝑥

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Figure drawn by course staff based on http://dx.doi.org/10.1007/s00283-011-9256-x


OR gate

NOT gate shifter gadget

𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥

𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥

𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥

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split gadget

crossover gadget

NOT

shiftershifter

𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥

𝑥𝑥𝑥𝑥

𝑥𝑥

𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥

𝑥𝑥 𝑥𝑥

𝑣𝑣

𝑣𝑣𝑣𝑣

𝑣𝑣𝑣𝑣

𝑣𝑣𝑣𝑣

𝑣𝑣

𝑥𝑥 𝑥𝑥

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NAND

alternative crossover

gadget

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Figures 11 and 12 of http://simon.bailey.at/random/kaye.minesweeper.pdf which in turn is based on http://dx.doi.org/10.1145/1008354.1008356 Plus https://commons.wikimedia.org/wiki/File:NAND_from_NOR.svg#mediaviewer/File:NAND_from_NOR.svg


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Figure 7 of http://dx.doi.org/10.1007/s00283-011-9256-x

Functionally Complete Logic Gates

[Wernick 1942]

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http://en.wikipedia.org/wiki/Functional_completeness

Candy Crush is NP-complete[Walsh 2014]

variablegadget

true

false

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Figure drawn by course staff based on http://arxiv.org/abs/1403.1911


connectorgadgets

positive

negative

out

out

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wiregadget

out

out

in

in

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out

out

in

in

modifiedwire

gadget

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𝑥𝑥1

clausegadget

𝑥𝑥1

𝑥𝑥2

𝑥𝑥2

𝑥𝑥3

𝑥𝑥3

rewardgadget

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Bejeweled, Candy Crush, … areNP-Complete [Guala, Leucci, Natale 2014]

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http://arxiv.org/abs/1403.5830 Play the reduction: http://candycrush.isnphard.com/

super mario bros. is np-hardcourses.csail.mit.edu/6.892/spring19/lectures/l06_images.pdf ·...

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