icosian game [hamilton 1857] opponents form obstacles settlers of catan mate-in-1 is np-complete...

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  • Icosian Game [Hamilton 1857]

    Presenter Presentation Notes http://puzzlemuseum.com/month/picm02/200207icosian.htm

  • 𝑎𝑎 ∨ 𝑏𝑏 ∨ 𝑐𝑐

    clause

    variable

    Planar (Directed) Hamiltonian Cycle [Lichtenstein 1982]

    Presenter Presentation Notes http://dx.doi.org/10.1137/0211025

  • 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

  • Planar Directed Max-Degree-3 [Plesńik 1979]

    Presenter Presentation Notes http://www.aya.or.jp/~babalabo/DownLoad/Plesnik%208.4.192-196.pdf

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

    Presenter Presentation Notes http://dx.doi.org/10.1137/0211056

  • 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

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

    scale 3×

    Presenter Presentation Notes http://dx.doi.org/10.1137/0211056

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

    edge gadget

    Presenter Presentation Notes http://dx.doi.org/10.1137/0211056

  • vertex gadget

    vertex-edge connections

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

    Presenter Presentation Notes http://dx.doi.org/10.1137/0211056

  • used edge gadget

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

    unused edge gadget

    Presenter Presentation Notes http://dx.doi.org/10.1137/0211056

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

    Presenter Presentation Notes http://dx.doi.org/10.1137/0211056

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

    edge gadget

    Presenter Presentation Notes http://dx.doi.org/10.1016/0196-6774(84)90029-4

  • 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

  • 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

  • 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

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

    Hamiltonicity in Triangular Grid Graphs

    Presenter Presentation Notes http://dx.doi.org/10.1016/j.comgeo.2008.11.004

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

    vertex-edge connections

    vertex gadget

    edge gadget

    Presenter Presentation Notes http://dx.doi.org/10.1016/j.comgeo.2008.11.004

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

    vertex-edge connections

    turn gadget

    Presenter Presentation Notes http://dx.doi.org/10.1016/j.comgeo.2008.11.004

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

    Hamiltonicity in Hexagonal Grid Graphs

    Presenter Presentation Notes http://dx.doi.org/10.1016/j.comgeo.2008.11.004

  • 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

  • Arkin, Fekete, Islam, Meijer,

    Mitchell, Núñez- Rodríguez, Polishchuk,

    Rappaport, Xiao 2009

    U-turn turn gadget vertex core

    Presenter Presentation Notes http://dx.doi.org/10.1016/j.comgeo.2008.11.004

  • 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

  • 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

  • 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

  • opponents form obstacles

    Settlers of Catan Mate-in-1 is NP-Complete [Demaine, van Eycke, McKay 2011] • Reduction from Hamiltonicity in hex grids

    [Arkin, Fekete, Islam, Meijer, Mitchell, Núñez-Rodríguez, Polishchuk, Rappaport, Xiao 2008]

    2 VP to win

    × ∞ (buys roads)

    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

  • Settlers of Catan Mate-in-0 is NP-Complete [Demaine, van Eycke, McKay 2011] • Reduction from Hamiltonicity in hex grids

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

    2 VP to 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

  • Slitherlink [Nikoli 1989]

    Presenter Presentation Notes Sample problem 1 from http://www.nikoli.com/en/puzzles/slitherlink/

  • Slitherlink is NP-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

  • Slitherlink is NP-complete

    [Yato 2000]

    Presenter Presentation Notes http://www-imai.is.s.u-tokyo.ac.jp/~yato/data2/SIGAL74-3.pdf

  • Hashiwokakero [Nikoli 1990]

    Presenter Presentation Notes Sample problem 1 from http://www.nikoli.com/en/puzzles/hashiwokakero/

  • Hashiwokakero is NP-Complete [Andersson 2009]

    Presenter Presentation Notes http://dx.doi.org/10.1016/j.ipl.2009.07.017

  • Lawn Mowing

    Presenter Presentation Notes https://play.google.com/store/apps/details?id=com.donutgames.sundaylawn

  • Laser Cutting

    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

  • 3D Printing

  • 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

  • Hamiltonicity in Unit Orthogonal Segment Intersection Graphs

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

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

  • Minimum-Turn Milling

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

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

    Icosian Game [Hamilton 1857] Planar (Directed) Hamiltonian Cycle�[Lichtenstein 1982] Planar Directed Max-Degree-3 �[Plesńik 1979] Planar Directed Max-Degree-3 �[Plesńik 1979] Planar Bipartite Max-Degree-3 �[Itai, Papadimitriou, Szwarcfiter 1982] Grid Graphs�[Itai, Papadimitriou, Szwarcfiter 1982] Planar Bipartite Graph Drawing�[Itai, Papadimitriou, Szwarcfiter 1982] Hamiltonicity in Grid Graphs�[Itai, Papadimitriou, Szwarcfiter 1982] Hamiltonicity in Grid Graphs�[Itai, Papadimitriou, Szwarcfiter 1982] Hamiltonicity in Grid Graphs�[Itai, Papadimitriou, Szwarcfiter 1982] Hamiltonicity in Grid Graphs�[Itai, Papadimitriou, Szwarcfiter 1982] Max-Degree-3 Grid Graphs�[Papadimitriou & Vazirani 1984] Max-Degree-3 Grid Graphs�[Papadimitriou & Vazirani 1984] Max-Degree-3 Grid Graphs�[Papadimitriou & Vazirani 1984] Slide Number 15 Hamiltonicity in�Triangular Grid Graphs Slide Number 17 Slide Number 18 Hamiltonicity in�Hexagonal Grid Graphs Slide Number 20 Slide Number 21 Slide Number 22 Slide Number 23 Settlers of Catan [Klaus Teuber 1995] Settlers of Catan Mate-in-1 is�NP-Complete [Demaine, van Eycke, McKay 2011] Settlers of Catan Mate-in-0 is�NP-Complete [Demaine, van Eycke, McKay 2011] Slitherlink [Nikoli 1989] Slitherlink is�NP-complete Slitherlink is�NP-complete Hashiwokakero [Nikoli 1990] Hashiwokakero is NP-Complete�[Andersson 2009] Lawn Mowing Laser�Cutting