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DOI: 10.4230/LIPIcs.FUN.2018.4

Tracks from hell - when finding a proof may be easier than checking it

Authors: Matteo Almanza, Stefano Leucci, and Alessandro Panconesi

Published in: LIPIcs, Volume 100, 9th International Conference on Fun with Algorithms (FUN 2018)

Abstract

We consider the popular smartphone game Trainyard: a puzzle game that requires the player to lay down tracks in order to route colored trains from departure stations to suitable arrival stations. While it is already known [Almanza et al., FUN 2016] that the problem of finding a solution to a given Trainyard instance (i.e., game level) is NP-hard, determining the computational complexity of checking whether a candidate solution (i.e., a track layout) solves the level was left as an open problem. In this paper we prove that this verification problem is PSPACE-complete, thus implying that Trainyard players might not only have a hard time finding solutions to a given level, but they might even be unable to efficiently recognize them.

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Matteo Almanza, Stefano Leucci, and Alessandro Panconesi. Tracks from hell - when finding a proof may be easier than checking it. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 4:1-4:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{almanza_et_al:LIPIcs.FUN.2018.4,
  author =	{Almanza, Matteo and Leucci, Stefano and Panconesi, Alessandro},
  title =	{{Tracks from hell - when finding a proof may be easier than checking it}},
  booktitle =	{9th International Conference on Fun with Algorithms (FUN 2018)},
  pages =	{4:1--4:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-067-5},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{100},
  editor =	{Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2018.4},
  URN =		{urn:nbn:de:0030-drops-87954},
  doi =		{10.4230/LIPIcs.FUN.2018.4},
  annote =	{Keywords: puzzle games, solitaire games, Trainyard, verification}
}

Document

DOI: 10.4230/LIPIcs.FUN.2016.2

Trainyard is NP-hard

Authors: Matteo Almanza, Stefano Leucci, and Alessandro Panconesi

Published in: LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)

Abstract

Recently, due to the widespread diffusion of smart-phones, mobile puzzle games have experienced a huge increase in their popularity. A successful puzzle has to be both captivating and challenging, and it has been suggested that this features are somehow related to their computational complexity. Indeed, many puzzle games - such as Mah-Jongg, Sokoban, Candy Crush, and 2048, to name a few - are known to be NP-hard. In this paper we consider Trainyard: a popular mobile puzzle game whose goal is to get colored trains from their initial stations to suitable destination stations. We prove that the problem of determining whether there exists a solution to a given Trainyard level is NP. We also provide an implementation of our hardness reduction (see http://trainyard.isnphard.com).

Cite as

Matteo Almanza, Stefano Leucci, and Alessandro Panconesi. Trainyard is NP-hard. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 2:1-2:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{almanza_et_al:LIPIcs.FUN.2016.2,
  author =	{Almanza, Matteo and Leucci, Stefano and Panconesi, Alessandro},
  title =	{{Trainyard is NP-hard}},
  booktitle =	{8th International Conference on Fun with Algorithms (FUN 2016)},
  pages =	{2:1--2:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-005-7},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{49},
  editor =	{Demaine, Erik D. and Grandoni, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.2},
  URN =		{urn:nbn:de:0030-drops-58796},
  doi =		{10.4230/LIPIcs.FUN.2016.2},
  annote =	{Keywords: Complexity of Games, Trainyard}
}

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Tracks from hell - when finding a proof may be easier than checking it

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Trainyard is NP-hard

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