• 1 – 1: Decisions, decisions, decisions• 1 – 2: P, NP & Reductions• 1 – 3: SAT and 3SAT

• 2 – 1: Super Mario Bros.! • 2 – 2: More video games

• 3 – 1: Ref. bois

• Problem: `The Travelling Salesman’• Input:• Output:

• Problem: `The Travelling Salesman’• Input:• Output:

• Problem: `The Travelling Salesman’• Input:• Output:

• Problem: `The Travelling Salesman’• Input:• Output:

d

l

• Decision problems: only two outputs

YES NOor

• Decision problems: only two outputs

or

TRUE FALSEor

• Decision problems: only two outputs

or

or

1 0or

`Boolean Algebra’

• Information represented by strings:

• Boolean functions:

• Decision problem/Language (of f):

• Mathematically analyse `computing’ via a model of computation:

e.g. Turing Machines ™

http://www.win.tue.nl/~gwoegi/P-versus-NP.htm

Theorem: If 𝑃𝑃 = 𝑁𝑁𝑃𝑃, then for every 𝑁𝑁𝑃𝑃 language 𝐿𝐿, there exists a polynomial-time TM 𝐵𝐵 that on input 𝑥𝑥 ∈ 𝐿𝐿 outputs a certificate for 𝑥𝑥.

Yes/No problem = Search problem!

“As hard as every problem in X”

“As hard as every problem in 𝑿𝑿”

Moral: Reduce from known

Hard problem to yourproblem

*From Lecture 1 of E. Demaine. 6.890 Algorithmic Lower Bounds: Fun with Hardness Proofs. Fall 2014. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

< in finite time

SAT3SAT

CLIQUE

Linear programming

Prime factorisation

Register allocation

SudokuMinesweeper

Subset sumVertex cover

Graph colouringProtein structure prediction

Knapsack problem

Deadlock

Guarding art galleries

Sorting

Hamiltonian path

Jigsaw

Set packing

Steiner treeBattleships

Residency matching

clause literal

G. Aloupis, E.D. Demaine, A. Guo and G. Viglietta, “Classic Nintendo Games are (Computationally) Hard,” Theoretical Computer Science, volume 586, 2015, pages 135–160

E.D. Demaine, G. Viglietta, and A. Williams, “Super Mario Bros.Is Harder/ Easier Than We Thought,” Proceedings of the 8th International Conference on Fun with Algorithms (FUN 2016), La Maddalena, Italy, June 8–10, 2016, 13:1–13:14

Theorem: It is NP-hard to decide whether the goal is reachable from the start of a stage in generalisedSuper Mario Bros.

Start gadget

Finish gadget

Crossover gadget

Glitches GIFs. Available at: http://tasvideos.org/GameResources/NES/SuperMarioBros.html

Proof is resilient to

glitches

• Sprites and Maps from NES Maps: http://www.nesmaps.com/maps/SuperMarioBrothers/sprites/SuperMarioBrothersSprites.html

• G. Aloupis, E.D. Demaine, A. Guo and G. Viglietta, “Classic Nintendo Games are (Computationally) Hard,” Theoretical Computer Science, volume 586, 2015, pages 135–160

• S. Arora and B. Barak, “Computational Complexity: A Modern Approach,” Cambridge University Press, 2009

• R. Breuklelaar, E. D. Demaine, S. Hohenberger, H. J. Hoogeboom, W.A. Kosters, and D. Libon-Newell, “Tetris is Hard, Even to Approximate,” International Journal of Computational Geometry and Applications, volume 14, number 1–2, 2004, pages 41–68.

• R. Breukelaar, H. J. Hoogeboom,and W.A. Kosters, “Tetris is Hard, Made easy”. Technical Report 2003-9, Leiden Institute of Advanced Computer Science, Universiteit Leiden, 2003

• E. Demaine. 6.890 Algorithmic Lower Bounds: Fun with Hardness Proofs. Fall 2014. Massachusetts Institute of Technology: MIT OpenCourseWare, https://ocw.mit.edu. License: Creative Commons BY-NC-SA.

• E.D. Demaine, G. Viglietta, and A. Williams, “Super Mario Bros.Is Harder/ Easier Than We Thought,” Proceedings of the 8th International Conference on Fun with Algorithms (FUN 2016), La Maddalena, Italy, June 8–10, 2016, 13:1–13:14

• E.D. Demaine, J. Lockhart, and J. Lynch, “The Computational Complexity of Portal and Other 3D Video Games,” CoRR arXiv:1611.10319.

• R. Kaye, “Minesweeper is NP-complete,” The Mathematical Intelligence 22:9–15, 2000.• P\neq NP quote: Scott Aaronson, Shetl-Optimized Blog, “The Scientific Case for P\neq NP,”

https://www.scottaaronson.com/blog/?p=1720• The P vs. NP page: http://www.win.tue.nl/~gwoegi/P-versus-NP.htm

• Robert A. Hearn and Erik D. Demaine. Games, puzzles, and computation. CRC Press, 2009.

• S. Arora and B. Barak, “Computational Complexity: A Modern Approach,” Cambridge University Press, 2009