super mario bros. is np-hard

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)

[Bern & Hayes 1996]

Akari / Light Up [Nikoli 2001]

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

wire gadget

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

split/negationgadget

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

split gadget NOT gate

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

OR/XNOR gate

crossover gadget

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

¬𝑥𝑥 ∨ 𝑥𝑥 ∧ 𝑦𝑦 ∨ 𝑧𝑧

𝑥𝑥

𝑥𝑥 𝑥𝑥

𝑥𝑥𝑥𝑥′

𝑥𝑥′

𝑧𝑧

𝑧𝑧

𝑥𝑥′𝑥𝑥

𝑦𝑦′

𝑦𝑦

𝑦𝑦

∨

∧

∨

𝑥𝑥

𝑥𝑥

Minesweeper Consistency isNP-complete [Kaye 2000]

wire

wire terminator

split,NOT,turn

𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 𝑥𝑥𝑥𝑥

𝑥𝑥𝑥𝑥 𝑥𝑥 𝑥𝑥

𝑥𝑥𝑥𝑥

𝑥𝑥𝑥𝑥

𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥

𝑥𝑥 𝑥𝑥

𝑥𝑥𝑥𝑥𝑥𝑥

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

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

Minesweeper Consistency isNP-complete [Kaye 2000]

crossover

XOR

[Goldschlager 1977]

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

turn gadgetwire gadget terminator gadget

𝑥𝑥 𝑥𝑥

𝑥𝑥

𝑥𝑥

𝑥𝑥 𝑥𝑥

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

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

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

NAND

alternative crossover

gadget

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

Functionally Complete Logic Gates

[Wernick 1942]

Candy Crush is NP-complete[Walsh 2014]

variablegadget

true

false

Candy Crush is NP-complete[Walsh 2014]

connectorgadgets

positive

negative

out

out

Candy Crush is NP-complete[Walsh 2014]

wiregadget

out

out

in

in

Candy Crush is NP-complete[Walsh 2014]

out

out

in

in

modifiedwire

gadget

Candy Crush is NP-complete[Walsh 2014]

out

out

in

in

modifiedwire

gadget

Candy Crush is NP-complete[Walsh 2014]

out

out

in

in

modifiedwire

gadget

Candy Crush is NP-complete[Walsh 2014]

𝑥𝑥1

clausegadget

𝑥𝑥1

𝑥𝑥2

𝑥𝑥2

𝑥𝑥3

𝑥𝑥3

rewardgadget

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

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