16 Search Results for "Brunner, Josh"

Document
DOI: 10.4230/LIPIcs.FUN.2026.27
Completing the Complexity Classification of 2-Solo Chess: Knights and Kings Are Hard

Authors: Kolja Kühn and Wendy Yi

Published in: LIPIcs, Volume 366, 13th International Conference on Fun with Algorithms (FUN 2026)

Abstract
We extend the study of the 2-Solo Chess problem which was first introduced by Aravind, Misra, and Mittal in 2022. 2-Solo Chess is a single-player variant of chess in which the player must clear the board via captures such that only one piece remains, with each piece capturing at most twice. It is known that the problem is solvable in polynomial time for instances containing only pawns, while it becomes NP-complete for instances restricted to rooks, bishops, or queens. In this work, we complete the complexity classification by proving that 2-Solo Chess is NP-complete if the instance contains only knights or only kings.
Cite as

Kolja Kühn and Wendy Yi. Completing the Complexity Classification of 2-Solo Chess: Knights and Kings Are Hard. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 27:1-27:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{kuhn_et_al:LIPIcs.FUN.2026.27,
  author =	{K\"{u}hn, Kolja and Yi, Wendy},
  title =	{{Completing the Complexity Classification of 2-Solo Chess: Knights and Kings Are Hard}},
  booktitle =	{13th International Conference on Fun with Algorithms (FUN 2026)},
  pages =	{27:1--27:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-417-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{366},
  editor =	{Iacono, John},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2026.27},
  URN =		{urn:nbn:de:0030-drops-257464},
  doi =		{10.4230/LIPIcs.FUN.2026.27},
  annote =	{Keywords: Solo chess, puzzle games, board games, NP-completeness}
}
Document
DOI: 10.4230/LIPIcs.FUN.2026.29
Finding Shortest Walks in Kuru Kuru Kururin

Authors: Mickaël Laurent and Maher Mallem

Published in: LIPIcs, Volume 366, 13th International Conference on Fun with Algorithms (FUN 2026)

Abstract
This paper serves as a celebration of the twenty-fifth anniversary of Kuru Kuru Kururin. Although this video game is presented as a collection of two-dimensional puzzles based on rotation, it naturally invites players to complete its levels as quickly as possible. This has led to a surprisingly rich and challenging playing field to finding foremost temporal walks. In this work, we tackle this problem both in theory and in practice. First, we introduce a model for the game and provide an in-depth complexity analysis. Most notably, we show how each gameplay mechanic independently brings a layer of NP-hardness and/or co-NP-hardness. We also provide a pseudo-polynomial time algorithm for the general problem and identify several cases which can be solved in polynomial time. Along the way, we discuss connections to the more established framework of temporal graphs, both in the point model and the interval model. Then, we propose simple and flexible algorithmic techniques to reduce state space and guide the search, offering trade-offs between precision and computation speed in practice. These techniques were implemented and tested using a full recreation of the game physics and the levels from the original game. We demonstrate the efficiency of our framework in several settings - with or without taking damage, with or without unintended game mechanics - and relate empirical struggles which we encountered in practice to our complexity analysis. Our implementation is open source and fully available online, offering a novel and amusing setting to benchmark shortest path algorithms.
Cite as

Mickaël Laurent and Maher Mallem. Finding Shortest Walks in Kuru Kuru Kururin. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{laurent_et_al:LIPIcs.FUN.2026.29,
  author =	{Laurent, Micka\"{e}l and Mallem, Maher},
  title =	{{Finding Shortest Walks in Kuru Kuru Kururin}},
  booktitle =	{13th International Conference on Fun with Algorithms (FUN 2026)},
  pages =	{29:1--29:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-417-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{366},
  editor =	{Iacono, John},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2026.29},
  URN =		{urn:nbn:de:0030-drops-257480},
  doi =		{10.4230/LIPIcs.FUN.2026.29},
  annote =	{Keywords: Shortest path, Complexity}
}
Document
DOI: 10.4230/LIPIcs.FUN.2026.32
Tetris Is Hard with Just One Piece Type

Authors: MIT Hardness Group, Josh Brunner, Erik D. Demaine, Della Hendrickson, and Jeffery Li

Published in: LIPIcs, Volume 366, 13th International Conference on Fun with Algorithms (FUN 2026)

Abstract
We analyze the computational complexity of Tetris clearing (determining whether the player can clear an initial board using a given sequence of pieces) and survival (determining whether the player can avoid losing before placing all the given pieces in an initial board) when restricted to a single polyomino piece type. We prove, for any tetromino piece type P except for O, the NP-hardness of Tetris clearing and survival under the standard Super Rotation System (SRS), even when the input sequence consists of only a specified number of P pieces. These surprising results disprove a 23-year-old conjecture on the computational complexity of Tetris with only I pieces (although our result is only for a specific rotation system). As a corollary, we prove the NP-hardness of Tetris clearing when the sequence of pieces has to be able to be generated from a 7k-bag randomizer for any positive integer k ≥ 1. On the positive side, we give polynomial-time algorithms for Tetris clearing and survival when the input sequence consists of only dominoes, assuming a particular rotation model, solving a version of a 9-year-old open problem. Along the way, we give polynomial-time algorithms for Tetris clearing and survival with 1 × k pieces (for any fixed k), provided the top k-1 rows are initially empty, showing that our I NP-hardness result needs to have filled cells in the top three rows.
Cite as

MIT Hardness Group, Josh Brunner, Erik D. Demaine, Della Hendrickson, and Jeffery Li. Tetris Is Hard with Just One Piece Type. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 32:1-32:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{mithardnessgroup_et_al:LIPIcs.FUN.2026.32,
  author =	{MIT Hardness Group and Brunner, Josh and Demaine, Erik D. and Hendrickson, Della and Li, Jeffery},
  title =	{{Tetris Is Hard with Just One Piece Type}},
  booktitle =	{13th International Conference on Fun with Algorithms (FUN 2026)},
  pages =	{32:1--32:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-417-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{366},
  editor =	{Iacono, John},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2026.32},
  URN =		{urn:nbn:de:0030-drops-257515},
  doi =		{10.4230/LIPIcs.FUN.2026.32},
  annote =	{Keywords: complexity, hardness, video games, counting}
}
Document
DOI: 10.4230/LIPIcs.FUN.2026.40
An ASP-Completeness Framework for Dynasty Puzzles

Authors: Kosuke Susukita

Published in: LIPIcs, Volume 366, 13th International Conference on Fun with Algorithms (FUN 2026)

Abstract
A certain class of pencil-and-paper puzzles shares common rules: given a grid, certain cells must be shaded such that i) no two shaded cells are orthogonally adjacent, and ii) all unshaded cells are orthogonally connected. Such puzzles are sometimes referred to as "dynasty puzzles" within parts of the online puzzle community. We introduce a framework for proving the ASP-completeness (i.e., NP-complete under parsimonious reductions) of various dynasty puzzles. We apply this framework to seven specific dynasty puzzles - Akichiwake, Aquapelago, Ayeheya, Guide Arrow, Heyawake, Hitori, and Kurodoko. As a consequence, for given k solutions of any of these puzzles, deciding whether a distinct solution exists is NP-complete, and counting the number of solutions is #P-complete. Our results strengthen the known result of ASP-completeness for Heyawake and establish the ASP-completeness of the other six puzzles. The main idea is to reconstruct the reduction from the Tree-Residue Vertex-Breaking Problem (TRVB) to the Hamiltonian Cycle Problem introduced by MIT Hardness Group (2024). In our framework, the connectivity of the unshaded cells ensures the connectivity of the shaded cells, allowing the shaded cells to simulate TRVB, which is also an alternative representation of the Hamiltonian cycles under certain conditions.
Cite as

Kosuke Susukita. An ASP-Completeness Framework for Dynasty Puzzles. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 40:1-40:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{susukita:LIPIcs.FUN.2026.40,
  author =	{Susukita, Kosuke},
  title =	{{An ASP-Completeness Framework for Dynasty Puzzles}},
  booktitle =	{13th International Conference on Fun with Algorithms (FUN 2026)},
  pages =	{40:1--40:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-417-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{366},
  editor =	{Iacono, John},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2026.40},
  URN =		{urn:nbn:de:0030-drops-257596},
  doi =		{10.4230/LIPIcs.FUN.2026.40},
  annote =	{Keywords: ASP-completeness, pencil-and-paper puzzles, dynasty puzzles, Hitori, Kurodoko, Hamiltonian cycle, Tree-Residue Vertex-Breaking}
}
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.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
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.
Cite as

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.ISAAC.2024.51
Easier Ways to Prove Counting Hard: A Dichotomy for Generalized #SAT, Applied to Constraint Graphs

Authors: MIT Hardness Group, Josh Brunner, Erik D. Demaine, Jenny Diomidova, Timothy Gomez, Markus Hecher, Frederick Stock, and Zixiang Zhou

Published in: LIPIcs, Volume 322, 35th International Symposium on Algorithms and Computation (ISAAC 2024)

Abstract
To prove #P-hardness, a single-call reduction from #2SAT needs a clause gadget to have exactly the same number of solutions for all satisfying assignments - no matter how many and which literals satisfy the clause. In this paper, we relax this condition, making it easier to find #P-hardness reductions. Specifically, we introduce a framework called Generalized #SAT where each clause contributes a term to the total count of solutions based on a given function of the literals. For two-variable clauses (a natural generalization of #2SAT), we prove a dichotomy theorem characterizing when Generalized #SAT is in FP versus #P-complete. Equipped with these tools, we analyze the complexity of counting solutions to Constraint Graph Satisfiability (CGS), a framework previously used to prove NP-hardness (and PSPACE-hardness) of many puzzles and games. We prove CGS ASP-hard, meaning that there is a parsimonious reduction (with algorithmic bijection on solutions) from every NP search problem, which implies #P-completeness. Then we analyze CGS restricted to various subsets of features (vertex and edge types), and prove most of them either easy (in FP) or hard (#P-complete). Most of our results also apply to planar constraint graphs. CGS is thus a second powerful framework for proving problems #P-hard, with reductions requiring very few gadgets.
Cite as

MIT Hardness Group, Josh Brunner, Erik D. Demaine, Jenny Diomidova, Timothy Gomez, Markus Hecher, Frederick Stock, and Zixiang Zhou. Easier Ways to Prove Counting Hard: A Dichotomy for Generalized #SAT, Applied to Constraint Graphs. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mithardnessgroup_et_al:LIPIcs.ISAAC.2024.51,
  author =	{MIT Hardness Group and Brunner, Josh and Demaine, Erik D. and Diomidova, Jenny and Gomez, Timothy and Hecher, Markus and Stock, Frederick and Zhou, Zixiang},
  title =	{{Easier Ways to Prove Counting Hard: A Dichotomy for Generalized #SAT, Applied to Constraint Graphs}},
  booktitle =	{35th International Symposium on Algorithms and Computation (ISAAC 2024)},
  pages =	{51:1--51:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-354-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{322},
  editor =	{Mestre, Juli\'{a}n and Wirth, Anthony},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2024.51},
  URN =		{urn:nbn:de:0030-drops-221790},
  doi =		{10.4230/LIPIcs.ISAAC.2024.51},
  annote =	{Keywords: Counting, Computational Complexity, Sharp-P, Dichotomy, Constraint Graph Satisfiability}
}
Document
DOI: 10.4230/LIPIcs.SWAT.2024.34
Reconfiguration Algorithms for Cubic Modular Robots with Realistic Movement Constraints

Authors: MIT-NASA Space Robots Team, Josh Brunner, Kenneth C. Cheung, Erik D. Demaine, Jenny Diomidova, Christine Gregg, Della H. Hendrickson, and Irina Kostitsyna

Published in: LIPIcs, Volume 294, 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)

Abstract
We introduce and analyze a model for self-reconfigurable robots made up of unit-cube modules. Compared to past models, our model aims to newly capture two important practical aspects of real-world robots. First, modules often do not occupy an exact unit cube, but rather have features like bumps extending outside the allotted space so that modules can interlock. Thus, for example, our model forbids modules from squeezing in between two other modules that are one unit distance apart. Second, our model captures the practical scenario of many passive modules assembled by a single robot, instead of requiring all modules to be able to move on their own. We prove two universality results. First, with a supply of auxiliary modules, we show that any connected polycube structure can be constructed by a carefully aligned plane sweep. Second, without additional modules, we show how to construct any structure for which a natural notion of external feature size is at least a constant; this property largely consolidates forbidden-pattern properties used in previous works on reconfigurable modular robots.
Cite as

MIT-NASA Space Robots Team, Josh Brunner, Kenneth C. Cheung, Erik D. Demaine, Jenny Diomidova, Christine Gregg, Della H. Hendrickson, and Irina Kostitsyna. Reconfiguration Algorithms for Cubic Modular Robots with Realistic Movement Constraints. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 34:1-34:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{mitnasaspacerobotsteam_et_al:LIPIcs.SWAT.2024.34,
  author =	{MIT-NASA Space Robots Team and Brunner, Josh and Cheung, Kenneth C. and Demaine, Erik D. and Diomidova, Jenny and Gregg, Christine and Hendrickson, Della H. and Kostitsyna, Irina},
  title =	{{Reconfiguration Algorithms for Cubic Modular Robots with Realistic Movement Constraints}},
  booktitle =	{19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)},
  pages =	{34:1--34:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-318-8},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{294},
  editor =	{Bodlaender, Hans L.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.34},
  URN =		{urn:nbn:de:0030-drops-200742},
  doi =		{10.4230/LIPIcs.SWAT.2024.34},
  annote =	{Keywords: Modular robotics, programmable matter, digital materials, motion planning}
}
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.DNA.29.10
Complexity of Reconfiguration in Surface Chemical Reaction Networks

Authors: Robert M. Alaniz, Josh Brunner, Michael Coulombe, Erik D. Demaine, Jenny Diomidova, Timothy Gomez, Elise Grizzell, Ryan Knobel, Jayson Lynch, Andrew Rodriguez, Robert Schweller, and Tim Wylie

Published in: LIPIcs, Volume 276, 29th International Conference on DNA Computing and Molecular Programming (DNA 29) (2023)

Abstract
We analyze the computational complexity of basic reconfiguration problems for the recently introduced surface Chemical Reaction Networks (sCRNs), where ordered pairs of adjacent species nondeterministically transform into a different ordered pair of species according to a predefined set of allowed transition rules (chemical reactions). In particular, two questions that are fundamental to the simulation of sCRNs are whether a given configuration of molecules can ever transform into another given configuration, and whether a given cell can ever contain a given species, given a set of transition rules. We show that these problems can be solved in polynomial time, are NP-complete, or are PSPACE-complete in a variety of different settings, including when adjacent species just swap instead of arbitrary transformation (swap sCRNs), and when cells can change species a limited number of times (k-burnout). Most problems turn out to be at least NP-hard except with very few distinct species (2 or 3).
Cite as

Robert M. Alaniz, Josh Brunner, Michael Coulombe, Erik D. Demaine, Jenny Diomidova, Timothy Gomez, Elise Grizzell, Ryan Knobel, Jayson Lynch, Andrew Rodriguez, Robert Schweller, and Tim Wylie. Complexity of Reconfiguration in Surface Chemical Reaction Networks. In 29th International Conference on DNA Computing and Molecular Programming (DNA 29). Leibniz International Proceedings in Informatics (LIPIcs), Volume 276, pp. 10:1-10:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{alaniz_et_al:LIPIcs.DNA.29.10,
  author =	{Alaniz, Robert M. and Brunner, Josh and Coulombe, Michael and Demaine, Erik D. and Diomidova, Jenny and Gomez, Timothy and Grizzell, Elise and Knobel, Ryan and Lynch, Jayson and Rodriguez, Andrew and Schweller, Robert and Wylie, Tim},
  title =	{{Complexity of Reconfiguration in Surface Chemical Reaction Networks}},
  booktitle =	{29th International Conference on DNA Computing and Molecular Programming (DNA 29)},
  pages =	{10:1--10:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-297-6},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{276},
  editor =	{Chen, Ho-Lin and Evans, Constantine G.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DNA.29.10},
  URN =		{urn:nbn:de:0030-drops-187936},
  doi =		{10.4230/LIPIcs.DNA.29.10},
  annote =	{Keywords: Chemical Reaction Networks, reconfiguration, hardness}
}
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.ISAAC.2020.12
Arithmetic Expression Construction

Authors: Leo Alcock, Sualeh Asif, Jeffrey Bosboom, Josh Brunner, Charlotte Chen, Erik D. Demaine, Rogers Epstein, Adam Hesterberg, Lior Hirschfeld, William Hu, Jayson Lynch, Sarah Scheffler, and Lillian Zhang

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

Abstract
When can n given numbers be combined using arithmetic operators from a given subset of {+,-,×,÷} to obtain a given target number? We study three variations of this problem of Arithmetic Expression Construction: when the expression (1) is unconstrained; (2) has a specified pattern of parentheses and operators (and only the numbers need to be assigned to blanks); or (3) must match a specified ordering of the numbers (but the operators and parenthesization are free). For each of these variants, and many of the subsets of {+,-,×,÷}, we prove the problem NP-complete, sometimes in the weak sense and sometimes in the strong sense. Most of these proofs make use of a rational function framework which proves equivalence of these problems for values in rational functions with values in positive integers.
Cite as

Leo Alcock, Sualeh Asif, Jeffrey Bosboom, Josh Brunner, Charlotte Chen, Erik D. Demaine, Rogers Epstein, Adam Hesterberg, Lior Hirschfeld, William Hu, Jayson Lynch, Sarah Scheffler, and Lillian Zhang. Arithmetic Expression Construction. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{alcock_et_al:LIPIcs.ISAAC.2020.12,
  author =	{Alcock, Leo and Asif, Sualeh and Bosboom, Jeffrey and Brunner, Josh and Chen, Charlotte and Demaine, Erik D. and Epstein, Rogers and Hesterberg, Adam and Hirschfeld, Lior and Hu, William and Lynch, Jayson and Scheffler, Sarah and Zhang, Lillian},
  title =	{{Arithmetic Expression Construction}},
  booktitle =	{31st International Symposium on Algorithms and Computation (ISAAC 2020)},
  pages =	{12:1--12:15},
  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.2020.12},
  URN =		{urn:nbn:de:0030-drops-133568},
  doi =		{10.4230/LIPIcs.ISAAC.2020.12},
  annote =	{Keywords: Hardness, algebraic complexity, expression trees}
}
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.2021.7
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}
}
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