Presenter

Presentation Notes

http://courses.csail.mit.edu/6.892/spring19/

W. H. Freeman1979

A K PetersJuly 2009

translated byRyuhei Uehara

Complexity of Games & Puzzles

0 players(simulation)

1 player(puzzle)

2 players(game)

team,imperfect info

NP

PSPACE EXPTIME

P

Undecidable

NEXPTIMEPSPACE

PSPACE

Rengo Kriegspiel?

bridge?

Constraint Logic[Hearn & Demaine 2009]

PSPACE EXPTIME

P

Undecidable

NEXPTIME

PSPACE

NPPSPACE

0 players(simulation)

1 player(puzzle)

2 players(game)

team,imperfect info

Hamiltonian 𝒔𝒔, 𝒕𝒕 -Path

𝑣𝑣0 𝑣𝑣2 𝑣𝑣4

𝑣𝑣1

𝑣𝑣3

𝑠𝑠

𝑡𝑡𝑣𝑣0 𝑣𝑣2 𝑣𝑣4

𝑣𝑣1

𝑣𝑣3

𝑠𝑠

𝑡𝑡

input goal

100% Speedrun is NP-hard: MarioThe Lost Levels

Speedrun is NP-hard: Zelda

Speedrun is NP-hard: Metroidvania

Speedrun is NP-hard: RPG

Playing is NP-hard: Katamari

Edge-Matching Puzzles[1890s—]

photo by Matěj Baťha

[Christopher Monckton, 2007]US$2,000,000 prize (expired 2010)

NP-hardness of 𝟏𝟏 × 𝒏𝒏 Edge Matching

Reduction from Hamiltonian pathwith specified start 𝑠𝑠 & end vertex 𝑡𝑡

vertextile

edgetile

𝑣𝑣0 𝑣𝑣2 𝑣𝑣4

𝑣𝑣1

𝑣𝑣3

𝑠𝑠

𝑡𝑡

𝑠𝑠

𝑡𝑡

Bosboom, Demaine, Demaine, Hesterberg, Manurangsi, Yodpinyanee 2017

NP-hardness of 𝟏𝟏 × 𝒏𝒏 Edge Matching

or

𝐼𝐼𝑖𝑖 𝑂𝑂𝑖𝑖𝑈𝑈𝑖𝑖

𝑈𝑈𝑖𝑖𝐼𝐼𝑛𝑛𝑈𝑈𝑛𝑛

𝑈𝑈𝑛𝑛

𝑣𝑣𝑖𝑖 𝑡𝑡 = 𝑣𝑣𝑛𝑛𝑠𝑠 = 𝑣𝑣1

𝑣𝑣𝑖𝑖 ,𝑣𝑣𝑗𝑗 𝑣𝑣𝑖𝑖 ,𝑣𝑣𝑗𝑗

𝑂𝑂𝑖𝑖 𝐼𝐼𝑗𝑗𝑂𝑂𝑖𝑖

𝐼𝐼𝑗𝑗

NP-hardness

vertextile

edgetile

𝑣𝑣0 𝑣𝑣2 𝑣𝑣4

𝑣𝑣1

𝑣𝑣3

𝑠𝑠

𝑡𝑡

𝑠𝑠

𝑡𝑡

Bosboom, Demaine, Demaine, Hesterberg, Manurangsi, Yodpinyanee 2017