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Drainability and Fillability of Polyominoes in Diverse Models of Global Control

Authors Sándor P. Fekete , Peter Kramer , Jan-Marc Reinhardt , Christian Rieck , Christian Scheffer

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Global control
  • full Tilt
  • single Tilt
  • Fillability
  • Drainability
  • Polyominoes
  • Complexity

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Abstract

Tilt models offer intuitive and clean definitions of complex systems in which particles are influenced by global control commands. Despite a wide range of applications, there has been almost no theoretical investigation into the associated issues of filling and draining geometric environments. This is partly because a globally controlled system (i.e., passive matter) exhibits highly complex behavior that cannot be locally restricted. Thus, there is a strong need for theoretical studies that investigate these models both (1) in terms of relative power to each other, and (2) from a complexity theory perspective. In this work, we provide (1) general tools for comparing and contrasting different models of global control, and (2) both complexity and algorithmic results on filling and draining.

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Sándor P. Fekete, Peter Kramer, Jan-Marc Reinhardt, Christian Rieck, and Christian Scheffer. Drainability and Fillability of Polyominoes in Diverse Models of Global Control. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 74:1-74:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.74

Author Details

Sándor P. Fekete
  • Computer Science, TU Braunschweig, Germany
Peter Kramer
  • Computer Science, TU Braunschweig, Germany
Jan-Marc Reinhardt
  • Electrical Engineering and Computer Science, Bochum University of Applied Sciences, Germany
Christian Rieck
  • Discrete Mathematics, University of Kassel, Germany
Christian Scheffer
  • Electrical Engineering and Computer Science, Bochum University of Applied Sciences, Germany

Funding

Work from TU Braunschweig and HS Bochum was partially supported by the German Research Foundation (DFG), project "Space Ants", FE 407/22-1 and SCHE 1931/4-1. Work from University of Kassel was partially supported by DFG grant 522790373.

Acknowledgements

We are grateful for the detailed feedback provided in the reviews, which significantly improved the presentation of our paper. In particular, we thank one reviewer for pointing out [David Caballero et al., 2020] and its impact on Conjecture 31. We thank Erik D. Demaine for early helpful conversations.

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