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Icosian Game [Hamilton 1857]

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Page 1: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Icosian Game [Hamilton 1857]

Presenter
Presentation Notes
http://puzzlemuseum.com/month/picm02/200207icosian.htm
Page 2: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

𝑎𝑎 ∨ 𝑏𝑏 ∨ 𝑐𝑐

clause

variable

Planar (Directed) Hamiltonian Cycle[Lichtenstein 1982]

Presenter
Presentation Notes
http://dx.doi.org/10.1137/0211025
Page 3: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

𝑥𝑥1 ∨ 𝑥𝑥2∧ 𝑥𝑥1 ∨ 𝑥𝑥2 ∨ 𝑥𝑥3∧ 𝑥𝑥2 ∨ 𝑥𝑥3clause gadget

XOR gadget

Planar Directed Max-Degree-3 [Plesńik 1979]

Presenter
Presentation Notes
http://www.aya.or.jp/~babalabo/DownLoad/Plesnik%208.4.192-196.pdf
Page 4: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Planar Directed Max-Degree-3 [Plesńik 1979]

Presenter
Presentation Notes
http://www.aya.or.jp/~babalabo/DownLoad/Plesnik%208.4.192-196.pdf
Page 5: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Planar Bipartite Max-Degree-3 [Itai, Papadimitriou, Szwarcfiter 1982]

Presenter
Presentation Notes
http://dx.doi.org/10.1137/0211056
Page 6: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Grid Graphs[Itai, Papadimitriou, Szwarcfiter 1982]

figure by Arkin, Fekete, Islam, Meijer, Mitchell, Núñez-Rodríguez, Polishchuk, Rappaport, Xiao 2009

solid(no holes)

holes holes

Presenter
Presentation Notes
http://dx.doi.org/10.1137/0211056 Figure from http://dx.doi.org/10.1016/j.comgeo.2008.11.004
Page 7: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Planar Bipartite Graph Drawing[Itai, Papadimitriou, Szwarcfiter 1982]

scale 3×

Presenter
Presentation Notes
http://dx.doi.org/10.1137/0211056
Page 8: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Hamiltonicity in Grid Graphs[Itai, Papadimitriou, Szwarcfiter 1982]

edge gadget

Presenter
Presentation Notes
http://dx.doi.org/10.1137/0211056
Page 9: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

vertex gadget

vertex-edge connections

Hamiltonicity in Grid Graphs[Itai, Papadimitriou, Szwarcfiter 1982]

Presenter
Presentation Notes
http://dx.doi.org/10.1137/0211056
Page 10: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

used edge gadget

Hamiltonicity in Grid Graphs[Itai, Papadimitriou, Szwarcfiter 1982]

unused edge gadget

Presenter
Presentation Notes
http://dx.doi.org/10.1137/0211056
Page 11: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Hamiltonicity in Grid Graphs[Itai, Papadimitriou, Szwarcfiter 1982]

Presenter
Presentation Notes
http://dx.doi.org/10.1137/0211056
Page 12: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Max-Degree-3 Grid Graphs[Papadimitriou & Vazirani 1984]

edge gadget

Presenter
Presentation Notes
http://dx.doi.org/10.1016/0196-6774(84)90029-4
Page 13: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Max-Degree-3 Grid Graphs[Papadimitriou & Vazirani 1984]

vertex-edge connections

vertex gadget

Presenter
Presentation Notes
http://dx.doi.org/10.1016/0196-6774(84)90029-4
Page 14: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Max-Degree-3 Grid Graphs[Papadimitriou & Vazirani 1984]

forced-edge vertex-edge connections

Presenter
Presentation Notes
http://dx.doi.org/10.1016/0196-6774(84)90029-4
Page 15: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Arkin, Fekete, Islam, Meijer, Mitchell, Núñez-Rodríguez, Polishchuk, Rappaport, Xiao 2009

solid polygonal

Presenter
Presentation Notes
http://dx.doi.org/10.1016/j.comgeo.2008.11.004
Page 16: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Arkin, Fekete, Islam, Meijer, Mitchell, Núñez-Rodríguez, Polishchuk, Rappaport, Xiao 2009

Hamiltonicity inTriangular Grid Graphs

Presenter
Presentation Notes
http://dx.doi.org/10.1016/j.comgeo.2008.11.004
Page 17: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Arkin, Fekete, Islam, Meijer, Mitchell, Núñez-Rodríguez, Polishchuk, Rappaport, Xiao 2009

vertex-edgeconnections

vertex gadget

edgegadget

Presenter
Presentation Notes
http://dx.doi.org/10.1016/j.comgeo.2008.11.004
Page 18: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Arkin, Fekete, Islam, Meijer, Mitchell, Núñez-Rodríguez, Polishchuk, Rappaport, Xiao 2009

vertex-edgeconnections

turn gadget

Presenter
Presentation Notes
http://dx.doi.org/10.1016/j.comgeo.2008.11.004
Page 19: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Arkin, Fekete, Islam, Meijer, Mitchell, Núñez-Rodríguez, Polishchuk, Rappaport, Xiao 2009

Hamiltonicity inHexagonal Grid Graphs

Presenter
Presentation Notes
http://dx.doi.org/10.1016/j.comgeo.2008.11.004
Page 20: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Arkin, Fekete, Islam, Meijer, Mitchell, Núñez-Rodríguez, Polishchuk, Rappaport, Xiao 2009

edge gadget

Presenter
Presentation Notes
http://dx.doi.org/10.1016/j.comgeo.2008.11.004
Page 21: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Arkin, Fekete, Islam, Meijer,

Mitchell, Núñez-Rodríguez, Polishchuk,

Rappaport, Xiao 2009

U-turnturn gadget vertex core

Presenter
Presentation Notes
http://dx.doi.org/10.1016/j.comgeo.2008.11.004
Page 22: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Arkin, Fekete, Islam, Meijer, Mitchell, Núñez-Rodríguez, Polishchuk, Rappaport, Xiao 2009

vertex gadgets

Presenter
Presentation Notes
http://dx.doi.org/10.1016/j.comgeo.2008.11.004
Page 23: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Arkin, Fekete, Islam, Meijer, Mitchell, Núñez-Rodríguez, Polishchuk, Rappaport, Xiao 2009

Presenter
Presentation Notes
http://dx.doi.org/10.1016/j.comgeo.2008.11.004
Page 24: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Settlers of Catan [Klaus Teuber 1995]

> 15 million sold [Klaus Teuber 1995]

Presenter
Presentation Notes
http://www.catan.com/images/stories/Phase2Images/Settlers_of_Catan-p2.jpg from http://www.catan.com/catan-games/boardgame/basic-game.html
Page 25: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

opponentsformobstacles

Settlers of Catan Mate-in-1 isNP-Complete [Demaine, van Eycke, McKay 2011]

• Reduction from Hamiltonicity in hex grids[Arkin, Fekete, Islam, Meijer, Mitchell, Núñez-Rodríguez, Polishchuk, Rappaport, Xiao 2008]

2 VPto win

× ∞(buysroads)

opponent has road of 𝑛𝑛 − 1

Presenter
Presentation Notes
http://dx.doi.org/10.1016/j.comgeo.2008.11.004 http://pics.livejournal.com/greatgamegate/pic/000074qk from http://greatgamegate.livejournal.com/930.html
Page 26: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Settlers of Catan Mate-in-0 isNP-Complete [Demaine, van Eycke, McKay 2011]

• Reduction from Hamiltonicity in hex grids[Arkin, Fekete, Islam, Meijer, Mitchell, Núñez-Rodríguez, Polishchuk, Rappaport, Xiao 2009]

2 VPto win

In fact, deciding whether you’ve already won is NP-complete!

opponent has road of 𝑛𝑛 − 1

Presenter
Presentation Notes
http://dx.doi.org/10.1016/j.comgeo.2008.11.004 http://pics.livejournal.com/greatgamegate/pic/000074qk from http://greatgamegate.livejournal.com/930.html
Page 27: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Slitherlink [Nikoli 1989]

Presenter
Presentation Notes
Sample problem 1 from http://www.nikoli.com/en/puzzles/slitherlink/
Page 28: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Slitherlink isNP-complete

[Yato 2000]

optional vertex

required vertex

edgenon-edge

Presenter
Presentation Notes
http://www-imai.is.s.u-tokyo.ac.jp/~yato/data2/SIGAL74-3.pdf
Page 29: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Slitherlink isNP-complete

[Yato 2000]

Presenter
Presentation Notes
http://www-imai.is.s.u-tokyo.ac.jp/~yato/data2/SIGAL74-3.pdf
Page 30: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Hashiwokakero [Nikoli 1990]

Presenter
Presentation Notes
Sample problem 1 from http://www.nikoli.com/en/puzzles/hashiwokakero/
Page 31: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Hashiwokakero is NP-Complete[Andersson 2009]

Presenter
Presentation Notes
http://dx.doi.org/10.1016/j.ipl.2009.07.017
Page 32: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Lawn Mowing

Presenter
Presentation Notes
https://play.google.com/store/apps/details?id=com.donutgames.sundaylawn
Page 33: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

LaserCutting

George Hart at MIT,2003

Presenter
Presentation Notes
http://theory.csail.mit.edu/GeorgeHart/photos/prep/ from http://theory.csail.mit.edu/GeorgeHart/residence_101103.htm
Page 34: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

3D Printing

Page 35: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Milling & Lawn Mowing[Arkin, Fekete, Mitchell 2000]

Presenter
Presentation Notes
http://dx.doi.org/10.1016/S0925-7721(00)00015-8 http://www.ams.sunysb.edu/~estie/papers/lawn.pdf
Page 36: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Hamiltonicity in Unit Orthogonal Segment Intersection Graphs

[Arkin, Bender, Demaine, Fekete, Mitchell, Sethia 2005]

Presenter
Presentation Notes
http://erikdemaine.org/papers/Milling/
Page 37: Icosian Game [Hamilton 1857] …courses.csail.mit.edu/6.890/fall14/lectures/L08_images.pdfopponents form obstacles Settlers of Catan Mate-in-1 is NP-Complete [Demaine,van Eycke, McKay

Minimum-Turn Milling

[Arkin, Bender, Demaine, Fekete, Mitchell, Sethia 2005]

Presenter
Presentation Notes
http://erikdemaine.org/papers/Milling/

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