14 Search Results for "Hendrickson, Dylan"

Document
DOI: 10.4230/LIPIcs.ESA.2025.28
Sliding Squares in Parallel

Authors: Hugo A. Akitaya, Sándor P. Fekete, Peter Kramer, Saba Molaei, Christian Rieck, Frederick Stock, and Tobias Wallner

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We consider algorithmic problems motivated by modular robotic reconfiguration in the sliding square model, in which we are given n square-shaped modules in a (labeled or unlabeled) start configuration and need to find a schedule of sliding moves to transform it into a desired goal configuration, maintaining connectivity of the configuration at all times. Recent work has aimed at minimizing the total number of moves, resulting in fully sequential schedules that can perform reconfiguration in 𝒪(n²) moves, or 𝒪(nP) for arrangements of bounding box perimeter size P. We provide first results in the sliding square model that exploit parallel motion, performing reconfiguration in worst-case optimal makespan of 𝒪(P). We also provide tight bounds on the complexity of the problem by showing that even deciding the possibility of reconfiguration within makespan 1 is NP-complete in the unlabeled case. In the labeled variant, we note that deciding the same for makespan 2 is NP-complete, while makespan 1 is straightforward.
Cite as

Hugo A. Akitaya, Sándor P. Fekete, Peter Kramer, Saba Molaei, Christian Rieck, Frederick Stock, and Tobias Wallner. Sliding Squares in Parallel. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 28:1-28:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{a.akitaya_et_al:LIPIcs.ESA.2025.28,
  author =	{A. Akitaya, Hugo and Fekete, S\'{a}ndor P. and Kramer, Peter and Molaei, Saba and Rieck, Christian and Stock, Frederick and Wallner, Tobias},
  title =	{{Sliding Squares in Parallel}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{28:1--28:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.28},
  URN =		{urn:nbn:de:0030-drops-244961},
  doi =		{10.4230/LIPIcs.ESA.2025.28},
  annote =	{Keywords: Sliding squares, parallel motion, reconfigurability, motion planning, multi-agent path finding, makespan, swarm robotics, computational geometry}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.1
Reconfiguration of Unit Squares and Disks: PSPACE-Hardness in Simple Settings

Authors: Mikkel Abrahamsen, Kevin Buchin, Maike Buchin, Linda Kleist, Maarten Löffler, Lena Schlipf, André Schulz, and Jack Stade

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We study well-known reconfiguration problems. Given a start and a target configuration of geometric objects in a polygon, we wonder whether we can move the objects from the start configuration to the target configuration while avoiding collisions between the objects and staying within the polygon. Problems of this type have been considered since the early 80s by roboticists and computational geometers. In this paper, we study some of the simplest possible variants where the objects are labeled or unlabeled unit squares or unit disks. In unlabeled reconfiguration, the objects are identical, so that any object is allowed to end at any of the targets positions. In the labeled variant, each object has a designated target position. The results for the labeled variants are direct consequences from our insights on the unlabeled versions. We show that it is PSPACE-hard to decide whether there exists a reconfiguration of (unlabeled/labeled) unit squares even in a simple polygon. Previously, it was only known to be PSPACE-hard in a polygon with holes for both the unlabeled and labeled version [Solovey and Halperin, Int. J. Robotics Res. 2016]. Our proof is based on a result of independent interest, namely that reconfiguration between two satisfying assignments of a formula of Monotone-Planar-3-Sat is also PSPACE-complete. The reduction from reconfiguration of Monotone-Planar-3-Sat to reconfiguration of unit squares extends techniques recently developed to show NP-hardness of packing unit squares in a simple polygon [Abrahamsen and Stade, FOCS 2024]. We also show PSPACE-hardness of reconfiguration of (unlabeled/labeled) unit disks in a polygon with holes. Previously, it was known that unlabeled reconfiguration of disks of two different sizes was PSPACE-hard [Brocken, van der Heijden, Kostitsyna, Lo-Wong and Surtel, FUN 2021].
Cite as

Mikkel Abrahamsen, Kevin Buchin, Maike Buchin, Linda Kleist, Maarten Löffler, Lena Schlipf, André Schulz, and Jack Stade. Reconfiguration of Unit Squares and Disks: PSPACE-Hardness in Simple Settings. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2025.1,
  author =	{Abrahamsen, Mikkel and Buchin, Kevin and Buchin, Maike and Kleist, Linda and L\"{o}ffler, Maarten and Schlipf, Lena and Schulz, Andr\'{e} and Stade, Jack},
  title =	{{Reconfiguration of Unit Squares and Disks: PSPACE-Hardness in Simple Settings}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{1:1--1:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.1},
  URN =		{urn:nbn:de:0030-drops-231539},
  doi =		{10.4230/LIPIcs.SoCG.2025.1},
  annote =	{Keywords: reconfiguration, unit square, unit disk, unlabeled, labeled, simple polygon, polygon}
}
Document
Media Exposition
DOI: 10.4230/LIPIcs.SoCG.2025.85
Finding Shortest Reconfiguration Sequences for Modular Robots (Media Exposition)

Authors: UML Modular Robotics Group, Hugo A. Akitaya, Andrew Clements, Sam Downey, Jonathan Eisenbies, Soham Samanta, Gabriel Shahrouzi, and Frederick Stock

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
This paper introduces a set of tools built to help researchers design algorithms for modular robots. These tools can brute force solutions to specific reconfigurations, visualize movements of modular robots, and can be used to design specific configurations of robots. Multiple models of modular robots are supported, and can be added by users.
Cite as

UML Modular Robotics Group, Hugo A. Akitaya, Andrew Clements, Sam Downey, Jonathan Eisenbies, Soham Samanta, Gabriel Shahrouzi, and Frederick Stock. Finding Shortest Reconfiguration Sequences for Modular Robots (Media Exposition). In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 85:1-85:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{umlmodularroboticsgroup_et_al:LIPIcs.SoCG.2025.85,
  author =	{UML Modular Robotics Group and A. Akitaya, Hugo and Clements, Andrew and Downey, Sam and Eisenbies, Jonathan and Samanta, Soham and Shahrouzi, Gabriel and Stock, Frederick},
  title =	{{Finding Shortest Reconfiguration Sequences for Modular Robots}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{85:1--85:5},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.85},
  URN =		{urn:nbn:de:0030-drops-232371},
  doi =		{10.4230/LIPIcs.SoCG.2025.85},
  annote =	{Keywords: modular reconfigurable robots, sliding cube model, reconfiguration}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.5
Optimal Motion Planning for Two Square Robots in a Rectilinear Environment

Authors: Pankaj K. Agarwal, Mark de Berg, Benjamin Holmgren, Alex Steiger, and Martijn Struijs

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
Let W ⊂ ℝ² be a rectilinear polygonal environment (that is, a rectilinear polygon potentially with holes) with a total of n vertices, and let A,B be two robots, each modeled as an axis-aligned unit square, that can move rectilinearly inside W. The goal is to compute an optimal collision-free motion plan π for A and B between a given pair of source and target configurations. We study two variants of this problem and obtain the following results. - Min-Sum: Here the goal is to compute a motion plan that minimizes the sum of the lengths of the paths of the robots. We present an O(n⁴log n)-time algorithm for computing an optimal solution to the min-sum problem. This is the first polynomial-time algorithm to compute an optimal, collision-free motion of two robots amid obstacles in a planar polygonal environment. - Min-Makespan: Here the robots can move with at most unit speed, and the goal is to compute a motion plan that minimizes the maximum time taken by a robot to reach its target location. We prove that the min-makespan variant is NP-hard.
Cite as

Pankaj K. Agarwal, Mark de Berg, Benjamin Holmgren, Alex Steiger, and Martijn Struijs. Optimal Motion Planning for Two Square Robots in a Rectilinear Environment. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 5:1-5:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{agarwal_et_al:LIPIcs.SoCG.2025.5,
  author =	{Agarwal, Pankaj K. and de Berg, Mark and Holmgren, Benjamin and Steiger, Alex and Struijs, Martijn},
  title =	{{Optimal Motion Planning for Two Square Robots in a Rectilinear Environment}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{5:1--5:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.5},
  URN =		{urn:nbn:de:0030-drops-231577},
  doi =		{10.4230/LIPIcs.SoCG.2025.5},
  annote =	{Keywords: Computational geometry, motion planning, multiple robots, rectilinear paths}
}
Document
DOI: 10.4230/LIPIcs.SAND.2025.11
Hardness of Traversing Gadget Systems with Small Bandwidth

Authors: MIT Gadgets Group, Erik D. Demaine, Jenny Diomidova, Timothy Gomez, Markus Hecher, and Jayson Lynch

Published in: LIPIcs, Volume 330, 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)

Abstract
The motion-planning-through-gadgets framework has enabled proofs of PSPACE-completeness for many motion-planning problems, ranging from swarm and modular robotics to DNA computing to video games. In this paper, we strengthen this framework to show that, for several useful gadgets and gadget families, motion planning remains PSPACE-complete even when gadgets are connected together into a graph of constant bandwidth (which implies constant pathwidth, treewidth, and cliquewidth). We then show how this result applies to several geometric/grid-based motion-planning problems, establishing PSPACE-completeness even when restricted to a rectangle/box where only one dimension is large (superconstant). On the positive side, we find one family of gadgets (DAG gadgets) for which motion planning is fixed-parameter tractable with respect to bandwidth.
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MIT Gadgets Group, Erik D. Demaine, Jenny Diomidova, Timothy Gomez, Markus Hecher, and Jayson Lynch. Hardness of Traversing Gadget Systems with Small Bandwidth. In 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 330, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mitgadgetsgroup_et_al:LIPIcs.SAND.2025.11,
  author =	{MIT Gadgets Group and Demaine, Erik D. and Diomidova, Jenny and Gomez, Timothy and Hecher, Markus and Lynch, Jayson},
  title =	{{Hardness of Traversing Gadget Systems with Small Bandwidth}},
  booktitle =	{4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025)},
  pages =	{11:1--11:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-368-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{330},
  editor =	{Meeks, Kitty and Scheideler, Christian},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2025.11},
  URN =		{urn:nbn:de:0030-drops-230648},
  doi =		{10.4230/LIPIcs.SAND.2025.11},
  annote =	{Keywords: Gadgets, Motion Planning, Parameterized Complexity, Hardness}
}
Document
DOI: 10.4230/LIPIcs.SAND.2023.5
Complexity of Motion Planning of Arbitrarily Many Robots: Gadgets, Petri Nets, and Counter Machines

Authors: Joshua Ani, Michael Coulombe, Erik D. Demaine, Yevhenii Diomidov, Timothy Gomez, Dylan Hendrickson, and Jayson Lynch

Published in: LIPIcs, Volume 257, 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)

Abstract
We extend the motion-planning-through-gadgets framework to several new scenarios involving various numbers of robots/agents, and analyze the complexity of the resulting motion-planning problems. While past work considers just one robot or one robot per player, most of our models allow for one or more locations to spawn new robots in each time step, leading to arbitrarily many robots. In the 0-player context, where all motion is deterministically forced, we prove that deciding whether any robot ever reaches a specified location is undecidable, by representing a counter machine. In the 1-player context, where the player can choose how to move the robots, we prove equivalence to Petri nets, EXPSPACE-completeness for reaching a specified location, PSPACE-completeness for reconfiguration, and ACKERMANN-completeness for reconfiguration when robots can be destroyed in addition to spawned. Finally, we consider a variation on the standard 2-player context where, instead of one robot per player, we have one robot shared by the players, along with a ko rule to prevent immediately undoing the previous move. We prove this impartial 2-player game EXPTIME-complete.
Cite as

Joshua Ani, Michael Coulombe, Erik D. Demaine, Yevhenii Diomidov, Timothy Gomez, Dylan Hendrickson, and Jayson Lynch. Complexity of Motion Planning of Arbitrarily Many Robots: Gadgets, Petri Nets, and Counter Machines. In 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 5:1-5:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ani_et_al:LIPIcs.SAND.2023.5,
  author =	{Ani, Joshua and Coulombe, Michael and Demaine, Erik D. and Diomidov, Yevhenii and Gomez, Timothy and Hendrickson, Dylan and Lynch, Jayson},
  title =	{{Complexity of Motion Planning of Arbitrarily Many Robots: Gadgets, Petri Nets, and Counter Machines}},
  booktitle =	{2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023)},
  pages =	{5:1--5:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-275-4},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{257},
  editor =	{Doty, David and Spirakis, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SAND.2023.5},
  URN =		{urn:nbn:de:0030-drops-179414},
  doi =		{10.4230/LIPIcs.SAND.2023.5},
  annote =	{Keywords: Gadgets, robots, undecidability, Petri nets}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2022.32
Lower Bounds on Retroactive Data Structures

Authors: Lily Chung, Erik D. Demaine, Dylan Hendrickson, and Jayson Lynch

Published in: LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Abstract
We prove essentially optimal fine-grained lower bounds on the gap between a data structure and a partially retroactive version of the same data structure. Precisely, assuming any one of three standard conjectures, we describe a problem that has a data structure where operations run in O(T(n,m)) time per operation, but any partially retroactive version of that data structure requires T(n,m)⋅m^{1-o(1)} worst-case time per operation, where n is the size of the data structure at any time and m is the number of operations. Any data structure with operations running in O(T(n,m)) time per operation can be converted (via the "rollback method") into a partially retroactive data structure running in O(T(n,m)⋅m) time per operation, so our lower bound is tight up to an m^o(1) factor common in fine-grained complexity.
Cite as

Lily Chung, Erik D. Demaine, Dylan Hendrickson, and Jayson Lynch. Lower Bounds on Retroactive Data Structures. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 32:1-32:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chung_et_al:LIPIcs.ISAAC.2022.32,
  author =	{Chung, Lily and Demaine, Erik D. and Hendrickson, Dylan and Lynch, Jayson},
  title =	{{Lower Bounds on Retroactive Data Structures}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{32:1--32:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.32},
  URN =		{urn:nbn:de:0030-drops-173171},
  doi =		{10.4230/LIPIcs.ISAAC.2022.32},
  annote =	{Keywords: Retroactivity, time travel, rollback, fine-grained complexity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.29
Flat Folding an Unassigned Single-Vertex Complex (Combinatorially Embedded Planar Graph with Specified Edge Lengths) Without Flat Angles

Authors: Lily Chung, Erik D. Demaine, Dylan Hendrickson, and Victor Luo

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
A foundational result in origami mathematics is Kawasaki and Justin’s simple, efficient characterization of flat foldability for unassigned single-vertex crease patterns (where each crease can fold mountain or valley) on flat material. This result was later generalized to cones of material, where the angles glued at the single vertex may not sum to 360^∘. Here we generalize these results to when the material forms a complex (instead of a manifold), and thus the angles are glued at the single vertex in the structure of an arbitrary planar graph (instead of a cycle). Like the earlier characterizations, we require all creases to fold mountain or valley, not remain unfolded flat; otherwise, the problem is known to be NP-complete (weakly for flat material and strongly for complexes). Equivalently, we efficiently characterize which combinatorially embedded planar graphs with prescribed edge lengths can fold flat, when all angles must be mountain or valley (not unfolded flat). Our algorithm runs in O(n log³ n) time, improving on the previous best algorithm of O(n² log n).
Cite as

Lily Chung, Erik D. Demaine, Dylan Hendrickson, and Victor Luo. Flat Folding an Unassigned Single-Vertex Complex (Combinatorially Embedded Planar Graph with Specified Edge Lengths) Without Flat Angles. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 29:1-29:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chung_et_al:LIPIcs.SoCG.2022.29,
  author =	{Chung, Lily and Demaine, Erik D. and Hendrickson, Dylan and Luo, Victor},
  title =	{{Flat Folding an Unassigned Single-Vertex Complex (Combinatorially Embedded Planar Graph with Specified Edge Lengths) Without Flat Angles}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{29:1--29:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.29},
  URN =		{urn:nbn:de:0030-drops-160371},
  doi =		{10.4230/LIPIcs.SoCG.2022.29},
  annote =	{Keywords: Graph drawing, folding, origami, polyhedral complex, algorithms}
}
Document
DOI: 10.4230/LIPIcs.FUN.2022.3
Pushing Blocks via Checkable Gadgets: PSPACE-Completeness of Push-1F and Block/Box Dude

Authors: Joshua Ani, Lily Chung, Erik D. Demaine, Yevhenii Diomidov, Dylan Hendrickson, and Jayson Lynch

Published in: LIPIcs, Volume 226, 11th International Conference on Fun with Algorithms (FUN 2022)

Abstract
We prove PSPACE-completeness of the well-studied pushing-block puzzle Push-1F, a theoretical abstraction of many video games (first posed in 1999). We also prove PSPACE-completeness of two versions of the recently studied block-moving puzzle game with gravity, Block Dude - a video game dating back to 1994 - featuring either liftable blocks or pushable blocks. Two of our reductions are built on a new framework for "checkable" gadgets, extending the motion-planning-through-gadgets framework to support gadgets that can be misused, provided those misuses can be detected later.
Cite as

Joshua Ani, Lily Chung, Erik D. Demaine, Yevhenii Diomidov, Dylan Hendrickson, and Jayson Lynch. Pushing Blocks via Checkable Gadgets: PSPACE-Completeness of Push-1F and Block/Box Dude. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 3:1-3:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ani_et_al:LIPIcs.FUN.2022.3,
  author =	{Ani, Joshua and Chung, Lily and Demaine, Erik D. and Diomidov, Yevhenii and Hendrickson, Dylan and Lynch, Jayson},
  title =	{{Pushing Blocks via Checkable Gadgets: PSPACE-Completeness of Push-1F and Block/Box Dude}},
  booktitle =	{11th International Conference on Fun with Algorithms (FUN 2022)},
  pages =	{3:1--3:30},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-232-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{226},
  editor =	{Fraigniaud, Pierre and Uno, Yushi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2022.3},
  URN =		{urn:nbn:de:0030-drops-159737},
  doi =		{10.4230/LIPIcs.FUN.2022.3},
  annote =	{Keywords: gadgets, motion planning, hardness of games}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2021.10
Characterizing Universal Reconfigurability of Modular Pivoting Robots

Authors: Hugo A. Akitaya, Erik D. Demaine, Andrei Gonczi, Dylan H. Hendrickson, Adam Hesterberg, Matias Korman, Oliver Korten, Jayson Lynch, Irene Parada, and Vera Sacristán

Published in: LIPIcs, Volume 189, 37th International Symposium on Computational Geometry (SoCG 2021)

Abstract
We give both efficient algorithms and hardness results for reconfiguring between two connected configurations of modules in the hexagonal grid. The reconfiguration moves that we consider are "pivots", where a hexagonal module rotates around a vertex shared with another module. Following prior work on modular robots, we define two natural sets of hexagon pivoting moves of increasing power: restricted and monkey moves. When we allow both moves, we present the first universal reconfiguration algorithm, which transforms between any two connected configurations using O(n³) monkey moves. This result strongly contrasts the analogous problem for squares, where there are rigid examples that do not have a single pivoting move preserving connectivity. On the other hand, if we only allow restricted moves, we prove that the reconfiguration problem becomes PSPACE-complete. Moreover, we show that, in contrast to hexagons, the reconfiguration problem for pivoting squares is PSPACE-complete regardless of the set of pivoting moves allowed. In the process, we strengthen the reduction framework of Demaine et al. [FUN'18] that we consider of independent interest.
Cite as

Hugo A. Akitaya, Erik D. Demaine, Andrei Gonczi, Dylan H. Hendrickson, Adam Hesterberg, Matias Korman, Oliver Korten, Jayson Lynch, Irene Parada, and Vera Sacristán. Characterizing Universal Reconfigurability of Modular Pivoting Robots. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 10:1-10:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{a.akitaya_et_al:LIPIcs.SoCG.2021.10,
  author =	{A. Akitaya, Hugo and Demaine, Erik D. and Gonczi, Andrei and Hendrickson, Dylan H. and Hesterberg, Adam and Korman, Matias and Korten, Oliver and Lynch, Jayson and Parada, Irene and Sacrist\'{a}n, Vera},
  title =	{{Characterizing Universal Reconfigurability of Modular Pivoting Robots}},
  booktitle =	{37th International Symposium on Computational Geometry (SoCG 2021)},
  pages =	{10:1--10:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-184-9},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{189},
  editor =	{Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2021.10},
  URN =		{urn:nbn:de:0030-drops-138094},
  doi =		{10.4230/LIPIcs.SoCG.2021.10},
  annote =	{Keywords: reconfiguration, geometric algorithm, PSPACE-hardness, pivoting hexagons, pivoting squares, modular robots}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2020.17
Complexity of Retrograde and Helpmate Chess Problems: Even Cooperative Chess Is Hard

Authors: Josh Brunner, Erik D. Demaine, Dylan Hendrickson, and Julian Wellman

Published in: LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Abstract
We prove PSPACE-completeness of two classic types of Chess problems when generalized to n × n boards. A "retrograde" problem asks whether it is possible for a position to be reached from a natural starting position, i.e., whether the position is "valid" or "legal" or "reachable". Most real-world retrograde Chess problems ask for the last few moves of such a sequence; we analyze the decision question which gets at the existence of an exponentially long move sequence. A "helpmate" problem asks whether it is possible for a player to become checkmated by any sequence of moves from a given position. A helpmate problem is essentially a cooperative form of Chess, where both players work together to cause a particular player to win; it also arises in regular Chess games, where a player who runs out of time (flags) loses only if they could ever possibly be checkmated from the current position (i.e., the helpmate problem has a solution). Our PSPACE-hardness reductions are from a variant of a puzzle game called Subway Shuffle.
Cite as

Josh Brunner, Erik D. Demaine, Dylan Hendrickson, and Julian Wellman. Complexity of Retrograde and Helpmate Chess Problems: Even Cooperative Chess Is Hard. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{brunner_et_al:LIPIcs.ISAAC.2020.17,
  author =	{Brunner, Josh and Demaine, Erik D. and Hendrickson, Dylan and Wellman, Julian},
  title =	{{Complexity of Retrograde and Helpmate Chess Problems: Even Cooperative Chess Is Hard}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{17:1--17:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-173-3},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{181},
  editor =	{Cao, Yixin and Cheng, Siu-Wing and Li, Minming},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.17},
  URN =		{urn:nbn:de:0030-drops-133618},
  doi =		{10.4230/LIPIcs.ISAAC.2020.17},
  annote =	{Keywords: hardness, board games, PSPACE}
}
Document
DOI: 10.4230/LIPIcs.FUN.2021.3
Walking Through Doors Is Hard, Even Without Staircases: Proving PSPACE-Hardness via Planar Assemblies of Door Gadgets

Authors: Joshua Ani, Jeffrey Bosboom, Erik D. Demaine, Yenhenii Diomidov, Dylan Hendrickson, and Jayson Lynch

Published in: LIPIcs, Volume 157, 10th International Conference on Fun with Algorithms (FUN 2021) (2020)

Abstract
A door gadget has two states and three tunnels that can be traversed by an agent (player, robot, etc.): the "open" and "close" tunnel sets the gadget’s state to open and closed, respectively, while the "traverse" tunnel can be traversed if and only if the door is in the open state. We prove that it is PSPACE-complete to decide whether an agent can move from one location to another through a planar assembly of such door gadgets, removing the traditional need for crossover gadgets and thereby simplifying past PSPACE-hardness proofs of Lemmings and Nintendo games Super Mario Bros., Legend of Zelda, and Donkey Kong Country. Our result holds in all but one of the possible local planar embedding of the open, close, and traverse tunnels within a door gadget; in the one remaining case, we prove NP-hardness. We also introduce and analyze a simpler type of door gadget, called the self-closing door. This gadget has two states and only two tunnels, similar to the "open" and "traverse" tunnels of doors, except that traversing the traverse tunnel also closes the door. In a variant called the symmetric self-closing door, the "open" tunnel can be traversed if and only if the door is closed. We prove that it is PSPACE-complete to decide whether an agent can move from one location to another through a planar assembly of either type of self-closing door. Then we apply this framework to prove new PSPACE-hardness results for several 3D Mario games and Sokobond.
Cite as

Joshua Ani, Jeffrey Bosboom, Erik D. Demaine, Yenhenii Diomidov, Dylan Hendrickson, and Jayson Lynch. Walking Through Doors Is Hard, Even Without Staircases: Proving PSPACE-Hardness via Planar Assemblies of Door Gadgets. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 3:1-3:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{ani_et_al:LIPIcs.FUN.2021.3,
  author =	{Ani, Joshua and Bosboom, Jeffrey and Demaine, Erik D. and Diomidov, Yenhenii and Hendrickson, Dylan and Lynch, Jayson},
  title =	{{Walking Through Doors Is Hard, Even Without Staircases: Proving PSPACE-Hardness via Planar Assemblies of Door Gadgets}},
  booktitle =	{10th International Conference on Fun with Algorithms (FUN 2021)},
  pages =	{3:1--3:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-145-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{157},
  editor =	{Farach-Colton, Martin and Prencipe, Giuseppe and Uehara, Ryuhei},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2021.3},
  URN =		{urn:nbn:de:0030-drops-127642},
  doi =		{10.4230/LIPIcs.FUN.2021.3},
  annote =	{Keywords: gadgets, motion planning, hardness of games}
}
Document
DOI: 10.4230/LIPIcs.FUN.2021.7
1 X 1 Rush Hour with Fixed Blocks Is PSPACE-Complete

Authors: Josh Brunner, Lily Chung, Erik D. Demaine, Dylan Hendrickson, Adam Hesterberg, Adam Suhl, and Avi Zeff

Published in: LIPIcs, Volume 157, 10th International Conference on Fun with Algorithms (FUN 2021) (2020)

Abstract
Consider n²-1 unit-square blocks in an n × n square board, where each block is labeled as movable horizontally (only), movable vertically (only), or immovable - a variation of Rush Hour with only 1 × 1 cars and fixed blocks. We prove that it is PSPACE-complete to decide whether a given block can reach the left edge of the board, by reduction from Nondeterministic Constraint Logic via 2-color oriented Subway Shuffle. By contrast, polynomial-time algorithms are known for deciding whether a given block can be moved by one space, or when each block either is immovable or can move both horizontally and vertically. Our result answers a 15-year-old open problem by Tromp and Cilibrasi, and strengthens previous PSPACE-completeness results for Rush Hour with vertical 1 × 2 and horizontal 2 × 1 movable blocks and 4-color Subway Shuffle.
Cite as

Josh Brunner, Lily Chung, Erik D. Demaine, Dylan Hendrickson, Adam Hesterberg, Adam Suhl, and Avi Zeff. 1 X 1 Rush Hour with Fixed Blocks Is PSPACE-Complete. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{brunner_et_al:LIPIcs.FUN.2021.7,
  author =	{Brunner, Josh and Chung, Lily and Demaine, Erik D. and Hendrickson, Dylan and Hesterberg, Adam and Suhl, Adam and Zeff, Avi},
  title =	{{1 X 1 Rush Hour with Fixed Blocks Is PSPACE-Complete}},
  booktitle =	{10th International Conference on Fun with Algorithms (FUN 2021)},
  pages =	{7:1--7:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-145-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{157},
  editor =	{Farach-Colton, Martin and Prencipe, Giuseppe and Uehara, Ryuhei},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2021.7},
  URN =		{urn:nbn:de:0030-drops-127681},
  doi =		{10.4230/LIPIcs.FUN.2021.7},
  annote =	{Keywords: puzzles, sliding blocks, PSPACE-hardness}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2020.62
Toward a General Complexity Theory of Motion Planning: Characterizing Which Gadgets Make Games Hard

Authors: Erik D. Demaine, Dylan H. Hendrickson, and Jayson Lynch

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Abstract
We begin a general theory for characterizing the computational complexity of motion planning of robot(s) through a graph of "gadgets", where each gadget has its own state defining a set of allowed traversals which in turn modify the gadget’s state. We study two general families of such gadgets within this theory, one which naturally leads to motion planning problems with polynomially bounded solutions, and another which leads to polynomially unbounded (potentially exponential) solutions. We also study a range of competitive game-theoretic scenarios, from one player controlling one robot to teams of players each controlling their own robot and racing to achieve their team’s goal. Under certain restrictions on these gadgets, we fully characterize the complexity of bounded 1-player motion planning (NL vs. NP-complete), unbounded 1-player motion planning (NL vs. PSPACE-complete), and bounded 2-player motion planning (P vs. PSPACE-complete), and we partially characterize the complexity of unbounded 2-player motion planning (P vs. EXPTIME-complete), bounded 2-team motion planning (P vs. NEXPTIME-complete), and unbounded 2-team motion planning (P vs. undecidable). These results can be seen as an alternative to Constraint Logic (which has already proved useful as a basis for hardness reductions), providing a wide variety of agent-based gadgets, any one of which suffices to prove a problem hard.
Cite as

Erik D. Demaine, Dylan H. Hendrickson, and Jayson Lynch. Toward a General Complexity Theory of Motion Planning: Characterizing Which Gadgets Make Games Hard. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 62:1-62:42, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{demaine_et_al:LIPIcs.ITCS.2020.62,
  author =	{Demaine, Erik D. and Hendrickson, Dylan H. and Lynch, Jayson},
  title =	{{Toward a General Complexity Theory of Motion Planning: Characterizing Which Gadgets Make Games Hard}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{62:1--62:42},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.62},
  URN =		{urn:nbn:de:0030-drops-117478},
  doi =		{10.4230/LIPIcs.ITCS.2020.62},
  annote =	{Keywords: motion planning, computational complexity, NP, PSPACE, EXP, NEXP, undecidability, games}
}
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