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Documents authored by MIT Hardness Group

Document
DOI: 10.4230/LIPIcs.FUN.2024.21
PSPACE-Hard 2D Super Mario Games: Thirteen Doors

Authors: MIT Hardness Group, Hayashi Ani, Erik D. Demaine, Holden Hall, and Matias Korman

Published in: LIPIcs, Volume 291, 12th International Conference on Fun with Algorithms (FUN 2024)

Abstract
We prove PSPACE-hardness for fifteen games in the Super Mario Bros. 2D platforming video game series. Previously, only the original Super Mario Bros. was known to be PSPACE-hard (FUN 2016), though several of the games we study were known to be NP-hard (FUN 2014). Our reductions build door gadgets with open, close, and traverse traversals, in each case using mechanics unique to the game. While some of our door constructions are similar to those from FUN 2016, those for Super Mario Bros. 2, Super Mario Land 2, Super Mario World 2, and the New Super Mario Bros. series are quite different; notably, the Super Mario Bros. 2 door is extremely difficult. Doors remain elusive for just two 2D Mario games (Super Mario Land and Super Mario Run); we prove that these games are at least NP-hard.
Cite as

MIT Hardness Group, Hayashi Ani, Erik D. Demaine, Holden Hall, and Matias Korman. PSPACE-Hard 2D Super Mario Games: Thirteen Doors. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 21:1-21:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mithardnessgroup_et_al:LIPIcs.FUN.2024.21,
  author =	{MIT Hardness Group and Ani, Hayashi and Demaine, Erik D. and Hall, Holden and Korman, Matias},
  title =	{{PSPACE-Hard 2D Super Mario Games: Thirteen Doors}},
  booktitle =	{12th International Conference on Fun with Algorithms (FUN 2024)},
  pages =	{21:1--21:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-314-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{291},
  editor =	{Broder, Andrei Z. and Tamir, Tami},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.21},
  URN =		{urn:nbn:de:0030-drops-199295},
  doi =		{10.4230/LIPIcs.FUN.2024.21},
  annote =	{Keywords: video games, computational complexity, PSPACE}
}
Document
DOI: 10.4230/LIPIcs.FUN.2024.22
You Can't Solve These Super Mario Bros. Levels: Undecidable Mario Games

Authors: MIT Hardness Group, Hayashi Ani, Erik D. Demaine, Holden Hall, Ricardo Ruiz, and Naveen Venkat

Published in: LIPIcs, Volume 291, 12th International Conference on Fun with Algorithms (FUN 2024)

Abstract
We prove RE-completeness (and thus undecidability) of several 2D games in the Super Mario Bros. platform video game series: the New Super Mario Bros. series (original, Wii, U, and 2), and both Super Mario Maker games in all five game styles (Super Mario Bros. 1 and 3, Super Mario World, New Super Mario Bros. U, and Super Mario 3D World). These results hold even when we restrict to constant-size levels and screens, but they do require generalizing to allow arbitrarily many enemies at each location and onscreen, as well as allowing for exponentially large (or no) timer. In our Super Mario Maker reductions, we work within the standard screen size and use the property that the game engine remembers offscreen objects that are global because they are supported by "global ground". To prove these Mario results, we build a new theory of counter gadgets in the motion-planning-through-gadgets framework, and provide a suite of simple gadgets for which reachability is RE-complete.
Cite as

MIT Hardness Group, Hayashi Ani, Erik D. Demaine, Holden Hall, Ricardo Ruiz, and Naveen Venkat. You Can't Solve These Super Mario Bros. Levels: Undecidable Mario Games. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 22:1-22:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mithardnessgroup_et_al:LIPIcs.FUN.2024.22,
  author =	{MIT Hardness Group and Ani, Hayashi and Demaine, Erik D. and Hall, Holden and Ruiz, Ricardo and Venkat, Naveen},
  title =	{{You Can't Solve These Super Mario Bros. Levels: Undecidable Mario Games}},
  booktitle =	{12th International Conference on Fun with Algorithms (FUN 2024)},
  pages =	{22:1--22:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-314-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{291},
  editor =	{Broder, Andrei Z. and Tamir, Tami},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.22},
  URN =		{urn:nbn:de:0030-drops-199302},
  doi =		{10.4230/LIPIcs.FUN.2024.22},
  annote =	{Keywords: video games, computational complexity, undecidability}
}
Document
DOI: 10.4230/LIPIcs.FUN.2024.23
ASP-Completeness of Hamiltonicity in Grid Graphs, with Applications to Loop Puzzles

Authors: MIT Hardness Group, Josh Brunner, Lily Chung, Erik D. Demaine, Della Hendrickson, and Andy Tockman

Published in: LIPIcs, Volume 291, 12th International Conference on Fun with Algorithms (FUN 2024)

Abstract
We prove that Hamiltonicity in maximum-degree-3 grid graphs (directed or undirected) is ASP-complete, i.e., it has a parsimonious reduction from every NP search problem (including a polynomial-time bijection between solutions). As a consequence, given k Hamiltonian cycles, it is NP-complete to find another; and counting Hamiltonian cycles is #P-complete. If we require the grid graph’s vertices to form a full m × n rectangle, then we show that Hamiltonicity remains ASP-complete if the edges are directed or if we allow removing some edges (whereas including all undirected edges is known to be easy). These results enable us to develop a stronger "T-metacell" framework for proving ASP-completeness of rectangular puzzles, which requires building just a single gadget representing a degree-3 grid-graph vertex. We apply this general theory to prove ASP-completeness of 37 pencil-and-paper puzzles where the goal is to draw a loop subject to given constraints: Slalom, Onsen-meguri, Mejilink, Detour, Tapa-Like Loop, Kouchoku, Icelom; Masyu, Yajilin, Nagareru, Castle Wall, Moon or Sun, Country Road, Geradeweg, Maxi Loop, Mid-loop, Balance Loop, Simple Loop, Haisu, Reflect Link, Linesweeper; Vertex/Touch Slitherlink, Dotchi-Loop, Ovotovata, Building Walk, Rail Pool, Disorderly Loop, Ant Mill, Koburin, Mukkonn Enn, Rassi Silai, (Crossing) Ichimaga, Tapa, Canal View, and Aqre. The last 13 of these puzzles were not even known to be NP-hard. Along the way, we prove ASP-completeness of some simple forms of Tree-Residue Vertex-Breaking (TRVB), including planar multigraphs with degree-6 breakable vertices, or with degree-4 breakable and degree-1 unbreakable vertices.
Cite as

MIT Hardness Group, Josh Brunner, Lily Chung, Erik D. Demaine, Della Hendrickson, and Andy Tockman. ASP-Completeness of Hamiltonicity in Grid Graphs, with Applications to Loop Puzzles. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 23:1-23:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mithardnessgroup_et_al:LIPIcs.FUN.2024.23,
  author =	{MIT Hardness Group and Brunner, Josh and Chung, Lily and Demaine, Erik D. and Hendrickson, Della and Tockman, Andy},
  title =	{{ASP-Completeness of Hamiltonicity in Grid Graphs, with Applications to Loop Puzzles}},
  booktitle =	{12th International Conference on Fun with Algorithms (FUN 2024)},
  pages =	{23:1--23:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-314-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{291},
  editor =	{Broder, Andrei Z. and Tamir, Tami},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.23},
  URN =		{urn:nbn:de:0030-drops-199314},
  doi =		{10.4230/LIPIcs.FUN.2024.23},
  annote =	{Keywords: pencil-and-paper puzzles, computational complexity, parsimony}
}
Document
DOI: 10.4230/LIPIcs.FUN.2024.24
Tetris with Few Piece Types

Authors: MIT Hardness Group, Erik D. Demaine, Holden Hall, and Jeffery Li

Published in: LIPIcs, Volume 291, 12th International Conference on Fun with Algorithms (FUN 2024)

Abstract
We prove NP-hardness and #P-hardness of Tetris clearing (clearing an initial board using a given sequence of pieces) with the Super Rotation System (SRS), even when the pieces are limited to any two of the seven Tetris piece types. This result is the first advance on a question posed twenty years ago: which piece sets are easy vs. hard? All previous Tetris NP-hardness proofs used five of the seven piece types. We also prove ASP-completeness of Tetris clearing, using three piece types, as well as versions of 3-Partition and Numerical 3-Dimensional Matching where all input integers are distinct. Finally, we prove NP-hardness of Tetris survival and clearing under the "hard drops only" and "20G" modes, using two piece types, improving on a previous "hard drops only" result that used five piece types.
Cite as

MIT Hardness Group, Erik D. Demaine, Holden Hall, and Jeffery Li. Tetris with Few Piece Types. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mithardnessgroup_et_al:LIPIcs.FUN.2024.24,
  author =	{MIT Hardness Group and Demaine, Erik D. and Hall, Holden and Li, Jeffery},
  title =	{{Tetris with Few Piece Types}},
  booktitle =	{12th International Conference on Fun with Algorithms (FUN 2024)},
  pages =	{24:1--24:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-314-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{291},
  editor =	{Broder, Andrei Z. and Tamir, Tami},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.24},
  URN =		{urn:nbn:de:0030-drops-199322},
  doi =		{10.4230/LIPIcs.FUN.2024.24},
  annote =	{Keywords: complexity, hardness, video games, counting}
}
Document
DOI: 10.4230/LIPIcs.FUN.2024.25
Complexity of Planar Graph Orientation Consistency, Promise-Inference, and Uniqueness, with Applications to Minesweeper Variants

Authors: MIT Hardness Group, Della Hendrickson, and Andy Tockman

Published in: LIPIcs, Volume 291, 12th International Conference on Fun with Algorithms (FUN 2024)

Abstract
We study three problems related to the computational complexity of the popular game Minesweeper. The first is consistency: given a set of clues, is there any arrangement of mines that satisfies it? This problem has been known to be NP-complete since 2000 [Kaye, 2000], but our framework proves it as a side effect. The second is inference: given a set of clues, is there any cell that the player can prove is safe? The coNP-completeness of this problem has been in the literature since 2011 [Scott et al., 2011], but we discovered a flaw that we believe is present in all published results, and we provide a fixed proof. Finally, the third is solvability: given the full state of a Minesweeper game, can the player win the game by safely clicking all non-mine cells? This problem has not yet been studied, and we prove that it is coNP-complete.
Cite as

MIT Hardness Group, Della Hendrickson, and Andy Tockman. Complexity of Planar Graph Orientation Consistency, Promise-Inference, and Uniqueness, with Applications to Minesweeper Variants. In 12th International Conference on Fun with Algorithms (FUN 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 291, pp. 25:1-25:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mithardnessgroup_et_al:LIPIcs.FUN.2024.25,
  author =	{MIT Hardness Group and Hendrickson, Della and Tockman, Andy},
  title =	{{Complexity of Planar Graph Orientation Consistency, Promise-Inference, and Uniqueness, with Applications to Minesweeper Variants}},
  booktitle =	{12th International Conference on Fun with Algorithms (FUN 2024)},
  pages =	{25:1--25:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-314-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{291},
  editor =	{Broder, Andrei Z. and Tamir, Tami},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2024.25},
  URN =		{urn:nbn:de:0030-drops-199335},
  doi =		{10.4230/LIPIcs.FUN.2024.25},
  annote =	{Keywords: NP, coNP, hardness, minesweeper, solvability, gadgets, simulation}
}
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