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Hexasort - the Complexity of Stacking Colors on Graphs

Authors Linus Klocker , Simon Dominik Fink

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Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Hexasort
  • offline color stacking on graphs
  • NP-complete
  • polynomial-time solvable
  • dynamic programming

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Abstract

Many popular puzzle and matching games have been analyzed through the lens of computational complexity. Prominent examples include Sudoku [Takayuki Yato and Takahiro Seta, 2003], Candy Crush [Luciano Gualà et al., 2014], and Flood-It [Fellows et al., 2018]. A common theme among these widely played games is that their generalized decision versions are NP-hard, which is often thought of as a source of their inherent difficulty and addictive appeal to human players. In this paper, we study a popular single-player stacking game commonly known as Hexasort. The game can be modelled as placing colored stacks onto the vertices of a graph, where adjacent stacks of the same color merge and vanish according to deterministic rules. We prove that Hexasort is NP-hard, even when restricted to single-color stacks and progressively more constrained classes of graphs, culminating in strong NP-hardness on trees of either bounded height or degree. Towards fixed-parameter tractable algorithms, we identify settings in which the problem becomes polynomial-time solvable and present dynamic programming algorithms.

Cite As Get BibTex

Linus Klocker and Simon Dominik Fink. Hexasort - the Complexity of Stacking Colors on Graphs. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 26:1-26:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.FUN.2026.26

Author Details

Linus Klocker
  • TU Wien, Austria
Simon Dominik Fink
  • Department of Algorithms and Complexity, TU Wien, Austria

Funding

Vienna Science and Technology Fund (WWTF) grant [10.47379/ICT22029] and Austrian Science Fund (FWF) grant [10.55776/Y1329]

References

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