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)
PresenterPresentation Noteshttp://arxiv.org/abs/1203.1895
[Bern & Hayes 1996]
PresenterPresentation NotesFigure 13.7 of GFALOP
Akari / Light Up [Nikoli 2001]
PresenterPresentation Noteshttp://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
PresenterPresentation Noteshttps://web.archive.org/web/20141218232731/http://people.cs.umass.edu/~mcphailb/papers/2005lightup.pdf
Akari / Light Up is NP-complete[McPhail 2005]
split/negationgadget
PresenterPresentation Noteshttps://web.archive.org/web/20141218232731/http://people.cs.umass.edu/~mcphailb/papers/2005lightup.pdf
Akari / Light Up is NP-complete[McPhail 2005]
split gadget NOT gate
PresenterPresentation Noteshttps://web.archive.org/web/20141218232731/http://people.cs.umass.edu/~mcphailb/papers/2005lightup.pdf
Akari / Light Up is NP-complete[McPhail 2005]
OR/XNOR gate
crossover gadget
PresenterPresentation Noteshttps://web.archive.org/web/20141218232731/http://people.cs.umass.edu/~mcphailb/papers/2005lightup.pdf
Akari / Light Up is NP-complete[McPhail 2005]
¬𝑥𝑥 ∨ 𝑥𝑥 ∧ 𝑦𝑦 ∨ 𝑧𝑧
𝑥𝑥
𝑥𝑥 𝑥𝑥
𝑥𝑥𝑥𝑥′
𝑥𝑥′
𝑧𝑧
𝑧𝑧
𝑥𝑥′𝑥𝑥
𝑦𝑦′
𝑦𝑦
𝑦𝑦
∨
∧
∨
PresenterPresentation Noteshttps://web.archive.org/web/20141218232731/http://people.cs.umass.edu/~mcphailb/papers/2005lightup.pdf
PresenterPresentation Noteshttp://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
PresenterPresentation NotesFigure drawn by course staff based on http://simon.bailey.at/random/kaye.minesweeper.pdf
Minesweeper Consistency isNP-complete [Kaye 2000]
NOT
Phase changer(2 NOTs)
𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥
𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥 𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥
𝑥𝑥 𝑥𝑥
𝑥𝑥 𝑥𝑥
PresenterPresentation NotesFigure drawn by course staff based on http://simon.bailey.at/random/kaye.minesweeper.pdf
Minesweeper Consistency isNP-complete [Kaye 2000]
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
PresenterPresentation NotesFigure drawn by course staff based on http://simon.bailey.at/random/kaye.minesweeper.pdf
Minesweeper Consistency isNP-complete [Kaye 2000]
crossover
XOR
[Goldschlager 1977]
PresenterPresentation NotesFigures 11 and 12 of http://simon.bailey.at/random/kaye.minesweeper.pdfwhich 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
PresenterPresentation NotesFigure drawn by course staff based on http://dx.doi.org/10.1007/s00283-011-9256-x
Minesweeper is CoNP-Complete[Scott, Stege, van Rooij 2011]
OR gate
NOT gate shifter gadget
𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥
𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥
𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥
PresenterPresentation NotesFigure drawn by course staff based on http://dx.doi.org/10.1007/s00283-011-9256-x
Minesweeper is CoNP-Complete[Scott, Stege, van Rooij 2011]
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
PresenterPresentation NotesFigure drawn by course staff based on http://dx.doi.org/10.1007/s00283-011-9256-x
Minesweeper is CoNP-Complete[Scott, Stege, van Rooij 2011]
NAND
alternative crossover
gadget
PresenterPresentation NotesFigures 11 and 12 of http://simon.bailey.at/random/kaye.minesweeper.pdfwhich in turn is based on http://dx.doi.org/10.1145/1008354.1008356Plus https://commons.wikimedia.org/wiki/File:NAND_from_NOR.svg#mediaviewer/File:NAND_from_NOR.svg
Minesweeper is CoNP-Complete[Scott, Stege, van Rooij 2011]
PresenterPresentation NotesFigure 7 of http://dx.doi.org/10.1007/s00283-011-9256-x
Functionally Complete Logic Gates
[Wernick 1942]
PresenterPresentation Noteshttp://en.wikipedia.org/wiki/Functional_completeness
Candy Crush is NP-complete[Walsh 2014]
variablegadget
true
false
PresenterPresentation NotesFigure drawn by course staff based on http://arxiv.org/abs/1403.1911
Candy Crush is NP-complete[Walsh 2014]
connectorgadgets
positive
negative
out
out
PresenterPresentation NotesFigure drawn by course staff based on http://arxiv.org/abs/1403.1911
Candy Crush is NP-complete[Walsh 2014]
wiregadget
out
out
in
in
PresenterPresentation NotesFigure drawn by course staff based on http://arxiv.org/abs/1403.1911
Candy Crush is NP-complete[Walsh 2014]
out
out
in
in
modifiedwire
gadget
PresenterPresentation NotesFigure drawn by course staff based on http://arxiv.org/abs/1403.1911
Candy Crush is NP-complete[Walsh 2014]
out
out
in
in
modifiedwire
gadget
PresenterPresentation NotesFigure drawn by course staff based on http://arxiv.org/abs/1403.1911
Candy Crush is NP-complete[Walsh 2014]
out
out
in
in
modifiedwire
gadget
PresenterPresentation NotesFigure drawn by course staff based on http://arxiv.org/abs/1403.1911
Candy Crush is NP-complete[Walsh 2014]
𝑥𝑥1
clausegadget
𝑥𝑥1
𝑥𝑥2
𝑥𝑥2
𝑥𝑥3
𝑥𝑥3
rewardgadget
PresenterPresentation NotesFigure drawn by course staff based on http://arxiv.org/abs/1403.1911
Bejeweled, Candy Crush, … areNP-Complete [Guala, Leucci, Natale 2014]
PresenterPresentation Noteshttp://arxiv.org/abs/1403.5830Play the reduction: http://candycrush.isnphard.com/
Bejeweled, Candy Crush, … areNP-Complete [Guala, Leucci, Natale 2014]
PresenterPresentation Noteshttp://arxiv.org/abs/1403.5830Play the reduction: http://candycrush.isnphard.com/
Super Mario Bros. is NP-Hard�[Aloupis, Demaine, Guo, Viglietta 2014]Slide Number 2Akari / Light Up [Nikoli 2001]Akari / Light Up is NP-complete�[McPhail 2005]Akari / Light Up is NP-complete�[McPhail 2005]Akari / Light Up is NP-complete�[McPhail 2005]Akari / Light Up is NP-complete�[McPhail 2005]Akari / Light Up is NP-complete�[McPhail 2005]Slide Number 9Minesweeper Consistency is�NP-complete [Kaye 2000]Minesweeper Consistency is�NP-complete [Kaye 2000]Minesweeper Consistency is�NP-complete [Kaye 2000]Minesweeper Consistency is�NP-complete [Kaye 2000]Minesweeper is CoNP-Complete�[Scott, Stege, van Rooij 2011]Minesweeper is CoNP-Complete�[Scott, Stege, van Rooij 2011]Minesweeper is CoNP-Complete�[Scott, Stege, van Rooij 2011]Minesweeper is CoNP-Complete�[Scott, Stege, van Rooij 2011]Minesweeper is CoNP-Complete�[Scott, Stege, van Rooij 2011]Functionally Complete Logic GatesCandy Crush is NP-complete�[Walsh 2014]Candy Crush is NP-complete�[Walsh 2014]Candy Crush is NP-complete�[Walsh 2014]Candy Crush is NP-complete�[Walsh 2014]Candy Crush is NP-complete�[Walsh 2014]Candy Crush is NP-complete�[Walsh 2014]Candy Crush is NP-complete�[Walsh 2014]Bejeweled, Candy Crush, … are�NP-Complete [Guala, Leucci, Natale 2014]Bejeweled, Candy Crush, … are�NP-Complete [Guala, Leucci, Natale 2014]