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

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).

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} }

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

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.

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} }

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

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.

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} }

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**Published in:** LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Given a graph where every vertex has exactly one labeled token, how can we most quickly execute a given permutation on the tokens? In (sequential) token swapping, the goal is to use the shortest possible sequence of swaps, each of which exchanges the tokens at the two endpoints of an edge of the graph. In parallel token swapping, the goal is to use the fewest rounds, each of which consists of one or more swaps on the edges of a matching. We prove that both of these problems remain NP-hard when the graph is restricted to be a tree.
These token swapping problems have been studied by disparate groups of researchers in discrete mathematics, theoretical computer science, robot motion planning, game theory, and engineering. Previous work establishes NP-completeness on general graphs (for both problems), constant-factor approximation algorithms, and some poly-time exact algorithms for simple graph classes such as cliques, stars, paths, and cycles. Sequential and parallel token swapping on trees were first studied over thirty years ago (as "sorting with a transposition tree") and over twenty-five years ago (as "routing permutations via matchings"), yet their complexities were previously unknown.
We also show limitations on approximation of sequential token swapping on trees: we identify a broad class of algorithms that encompass all three known polynomial-time algorithms that achieve the best known approximation factor (which is 2) and show that no such algorithm can achieve an approximation factor less than 2.

Oswin Aichholzer, Erik D. Demaine, Matias Korman, Anna Lubiw, Jayson Lynch, Zuzana Masárová, Mikhail Rudoy, Virginia Vassilevska Williams, and Nicole Wein. Hardness of Token Swapping on Trees. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{aichholzer_et_al:LIPIcs.ESA.2022.3, author = {Aichholzer, Oswin and Demaine, Erik D. and Korman, Matias and Lubiw, Anna and Lynch, Jayson and Mas\'{a}rov\'{a}, Zuzana and Rudoy, Mikhail and Vassilevska Williams, Virginia and Wein, Nicole}, title = {{Hardness of Token Swapping on Trees}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {3:1--3:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-247-1}, ISSN = {1868-8969}, year = {2022}, volume = {244}, editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva 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.2022.3}, URN = {urn:nbn:de:0030-drops-169413}, doi = {10.4230/LIPIcs.ESA.2022.3}, annote = {Keywords: Sorting, Token swapping, Trees, NP-hard, Approximation} }

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

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.

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} }

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**Published in:** LIPIcs, Volume 212, 32nd International Symposium on Algorithms and Computation (ISAAC 2021)

We present a linear space data structure for Dynamic Evaluation of k-CNF Boolean Formulas which achieves O(m^{1-1/k}) query and variable update time where m is the number of clauses in the formula and clauses are of size at most a constant k. Our algorithm is additionally able to count the total number of satisfied clauses. We then show how this data structure can be parallelized in the PRAM model to achieve O(log m) span (i.e. parallel time) and still O(m^{1-1/k}) work. This parallel algorithm works in the stronger Binary Fork model.
We then give a series of lower bounds on the problem including an average-case result showing the lower bounds hold even when the updates to the variables are chosen at random. Specifically, a reduction from k-Clique shows that dynamically counting the number of satisfied clauses takes time at least n^{(2ω-3)/6 √{2k} -1 -o(√k)}, where 2 ≤ ω < 2.38 is the matrix multiplication constant. We show the Combinatorial k-Clique Hypothesis implies a lower bound of m^{(1-k^{-1/2})(1-o(1))} which suggests our algorithm is close to optimal without involving Matrix Multiplication or new techniques. We next give an average-case reduction to k-clique showing the prior lower bounds hold even when the updates are chosen at random. We use our conditional lower bound to show any Binary Fork algorithm solving these problems requires at least Ω(log m) span, which is tight against our algorithm in this model. Finally, we give an unconditional linear space lower bound for Dynamic k-CNF Boolean Formula Evaluation.

Rathish Das, Andrea Lincoln, Jayson Lynch, and J. Ian Munro. Dynamic Boolean Formula Evaluation. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 61:1-61:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{das_et_al:LIPIcs.ISAAC.2021.61, author = {Das, Rathish and Lincoln, Andrea and Lynch, Jayson and Munro, J. Ian}, title = {{Dynamic Boolean Formula Evaluation}}, booktitle = {32nd International Symposium on Algorithms and Computation (ISAAC 2021)}, pages = {61:1--61:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-214-3}, ISSN = {1868-8969}, year = {2021}, volume = {212}, editor = {Ahn, Hee-Kap and Sadakane, Kunihiko}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2021.61}, URN = {urn:nbn:de:0030-drops-154945}, doi = {10.4230/LIPIcs.ISAAC.2021.61}, annote = {Keywords: Data Structures, SAT, Dynamic Algorithms, Boolean Formulas, Fine-grained Complexity, Parallel Algorithms} }

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

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.

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} }

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

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.

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} }

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

Recursed is a 2D puzzle platform video game featuring "treasure chests" that, when jumped into, instantiate a room that can later be exited (similar to function calls), optionally generating a "jar" that returns back to that room (similar to continuations). We prove that Recursed is RE-complete and thus undecidable (not recursive) by a reduction from the Post Correspondence Problem. Our reduction is "practical": the reduction from PCP results in fully playable levels that abide by all constraints governing levels (including the 15 × 20 room size) designed for the main game. Our reduction is also "efficient": a Turing machine can be simulated by a Recursed level whose size is linear in the encoding size of the Turing machine and whose solution length is polynomial in the running time of the Turing machine.

Erik D. Demaine, Justin Kopinsky, and Jayson Lynch. Recursed Is Not Recursive: A Jarring Result. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 50:1-50:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{demaine_et_al:LIPIcs.ISAAC.2020.50, author = {Demaine, Erik D. and Kopinsky, Justin and Lynch, Jayson}, title = {{Recursed Is Not Recursive: A Jarring Result}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {50:1--50:15}, 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.2020.50}, URN = {urn:nbn:de:0030-drops-133940}, doi = {10.4230/LIPIcs.ISAAC.2020.50}, annote = {Keywords: Computational Complexity, Undecidable, Video Games} }

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

In the Nikoli pencil-and-paper game Tatamibari, a puzzle consists of an m x n grid of cells, where each cell possibly contains a clue among ⊞, ⊟, ◫. The goal is to partition the grid into disjoint rectangles, where every rectangle contains exactly one clue, rectangles containing ⊞ are square, rectangles containing ⊟ are strictly longer horizontally than vertically, rectangles containing ◫ are strictly longer vertically than horizontally, and no four rectangles share a corner. We prove this puzzle NP-complete, establishing a Nikoli gap of 16 years. Along the way, we introduce a gadget framework for proving hardness of similar puzzles involving area coverage, and show that it applies to an existing NP-hardness proof for Spiral Galaxies. We also present a mathematical puzzle font for Tatamibari.

Aviv Adler, Jeffrey Bosboom, Erik D. Demaine, Martin L. Demaine, Quanquan C. Liu, and Jayson Lynch. Tatamibari Is NP-Complete. In 10th International Conference on Fun with Algorithms (FUN 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 157, pp. 1:1-1:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{adler_et_al:LIPIcs.FUN.2021.1, author = {Adler, Aviv and Bosboom, Jeffrey and Demaine, Erik D. and Demaine, Martin L. and Liu, Quanquan C. and Lynch, Jayson}, title = {{Tatamibari Is NP-Complete}}, booktitle = {10th International Conference on Fun with Algorithms (FUN 2021)}, pages = {1:1--1:24}, 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.2021.1}, URN = {urn:nbn:de:0030-drops-127624}, doi = {10.4230/LIPIcs.FUN.2021.1}, annote = {Keywords: Nikoli puzzles, NP-hardness, rectangle covering} }

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

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.

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} }

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

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.

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

We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete ("witnesses exist but are hard to find"), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding Sigma_2-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies.

Zachary Abel, Jeffrey Bosboom, Erik D. Demaine, Linus Hamilton, Adam Hesterberg, Justin Kopinsky, Jayson Lynch, and Mikhail Rudoy. Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 3:1-3:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{abel_et_al:LIPIcs.FUN.2018.3, author = {Abel, Zachary and Bosboom, Jeffrey and Demaine, Erik D. and Hamilton, Linus and Hesterberg, Adam and Kopinsky, Justin and Lynch, Jayson and Rudoy, Mikhail}, title = {{Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {3:1--3:21}, 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.2018.3}, URN = {urn:nbn:de:0030-drops-87944}, doi = {10.4230/LIPIcs.FUN.2018.3}, annote = {Keywords: video games, puzzles, hardness} }

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

We show the undecidability of whether a team has a forced win in a number of well known video games including: Team Fortress 2, Super Smash Brothers: Brawl, and Mario Kart.To do so, we give a simplification of the Team Computation Game [Hearn and Demaine, 2009] and use that to give an undecidable abstract game on graphs. This graph game framework better captures the geometry and common constraints in many games and is thus a powerful tool for showing their computational complexity.

Michael J. Coulombe and Jayson Lynch. Cooperating in Video Games? Impossible! Undecidability of Team Multiplayer Games. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 14:1-14:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{coulombe_et_al:LIPIcs.FUN.2018.14, author = {Coulombe, Michael J. and Lynch, Jayson}, title = {{Cooperating in Video Games? Impossible! Undecidability of Team Multiplayer Games}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {14:1--14:16}, 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.2018.14}, URN = {urn:nbn:de:0030-drops-88051}, doi = {10.4230/LIPIcs.FUN.2018.14}, annote = {Keywords: computational complexity, undecidable, team games, imperfect information} }

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

We initiate a general theory for analyzing the complexity of motion planning of a single robot through a graph of "gadgets", each with their own state, set of locations, and allowed traversals between locations that can depend on and change the state. This type of setup is common to many robot motion planning hardness proofs. We characterize the complexity for a natural simple case: each gadget connects up to four locations in a perfect matching (but each direction can be traversable or not in the current state), has one or two states, every gadget traversal is immediately undoable, and that gadget locations are connected by an always-traversable forest, possibly restricted to avoid crossings in the plane. Specifically, we show that any single nontrivial four-location two-state gadget type is enough for motion planning to become PSPACE-complete, while any set of simpler gadgets (effectively two-location or one-state) has a polynomial-time motion planning algorithm. As a sample application, our results show that motion planning games with "spinners" are PSPACE-complete, establishing a new hard aspect of Zelda: Oracle of Seasons.

Erik D. Demaine, Isaac Grosof, Jayson Lynch, and Mikhail Rudoy. Computational Complexity of Motion Planning of a Robot through Simple Gadgets. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 18:1-18:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{demaine_et_al:LIPIcs.FUN.2018.18, author = {Demaine, Erik D. and Grosof, Isaac and Lynch, Jayson and Rudoy, Mikhail}, title = {{Computational Complexity of Motion Planning of a Robot through Simple Gadgets}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {18:1--18:21}, 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.2018.18}, URN = {urn:nbn:de:0030-drops-88098}, doi = {10.4230/LIPIcs.FUN.2018.18}, annote = {Keywords: PSPACE, hardness, motion planning, puzzles} }

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

We classify the computational complexity of the popular video games Portal and Portal 2. We isolate individual mechanics of the game and prove NP-hardness, PSPACE-completeness, or pseudo-polynomiality depending on the specific game mechanics allowed. One of our proofs generalizes to prove NP-hardness of many other video games such as Half-Life 2, Halo, Doom, Elder Scrolls, Fallout, Grand Theft Auto, Left 4 Dead, Mass Effect, Deus Ex, Metal Gear Solid, and Resident Evil. These results build on the established literature on the complexity of video games [Aloupis et al., 2014][Cormode, 2004][Forisek, 2010][Viglietta, 2014].

Erik D. Demaine, Joshua Lockhart, and Jayson Lynch. The Computational Complexity of Portal and Other 3D Video Games. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 19:1-19:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{demaine_et_al:LIPIcs.FUN.2018.19, author = {Demaine, Erik D. and Lockhart, Joshua and Lynch, Jayson}, title = {{The Computational Complexity of Portal and Other 3D Video Games}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {19:1--19:22}, 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.2018.19}, URN = {urn:nbn:de:0030-drops-88104}, doi = {10.4230/LIPIcs.FUN.2018.19}, annote = {Keywords: video games, hardness, motion planning, NP, PSPACE} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

This paper initiates the study of I/O algorithms (minimizing cache misses) from the perspective of fine-grained complexity
(conditional polynomial lower bounds). Specifically, we aim to answer why sparse graph problems are so hard, and why the Longest Common Subsequence problem gets a savings of a factor of the size of cache times the length of a cache line, but no more. We take the reductions and techniques from complexity and fine-grained complexity and apply them to the I/O model to generate new (conditional) lower bounds as well as new faster algorithms. We also prove the existence of a time hierarchy for the I/O model, which motivates the fine-grained reductions.
- Using fine-grained reductions, we give an algorithm for distinguishing 2 vs. 3 diameter and radius that runs in O(|E|^2/(MB)) cache misses, which for sparse graphs improves over the previous O(|V|^2/B) running time.
- We give new reductions from radius and diameter to Wiener index and median. These reductions are new in both the RAM and I/O models.
- We show meaningful reductions between problems that have linear-time solutions in the RAM model. The reductions use low I/O complexity (typically O(n/B)), and thus help to finely capture between "I/O linear time" O(n/B) and RAM linear time O(n).
- We generate new I/O assumptions based on the difficulty of improving sparse graph problem running times in the I/O model. We create conjectures that the current best known algorithms for Single Source Shortest Paths (SSSP), diameter, and radius are optimal.
- From these I/O-model assumptions, we show that many of the known reductions in the word-RAM model can naturally extend to hold in the I/O model as well (e.g., a lower bound on the I/O complexity of Longest Common Subsequence that matches the best known running time).
- We prove an analog of the Time Hierarchy Theorem in the I/O model, further motivating the study of fine-grained algorithmic differences.

Erik D. Demaine, Andrea Lincoln, Quanquan C. Liu, Jayson Lynch, and Virginia Vassilevska Williams. Fine-grained I/O Complexity via Reductions: New Lower Bounds, Faster Algorithms, and a Time Hierarchy. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 34:1-34:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{demaine_et_al:LIPIcs.ITCS.2018.34, author = {Demaine, Erik D. and Lincoln, Andrea and Liu, Quanquan C. and Lynch, Jayson and Vassilevska Williams, Virginia}, title = {{Fine-grained I/O Complexity via Reductions: New Lower Bounds, Faster Algorithms, and a Time Hierarchy}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {34:1--34:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.34}, URN = {urn:nbn:de:0030-drops-83335}, doi = {10.4230/LIPIcs.ITCS.2018.34}, annote = {Keywords: IO model, Fine-grained Complexity, Algorithms} }

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**Published in:** LIPIcs, Volume 51, 32nd International Symposium on Computational Geometry (SoCG 2016)

We exactly settle the complexity of graph realization, graph rigidity, and graph global rigidity as applied to three types of graphs: "globally noncrossing" graphs, which avoid crossings in all of their configurations; matchstick graphs, with unit-length edges and where only noncrossing configurations are considered; and unrestricted graphs (crossings allowed) with unit edge lengths (or in the global rigidity case, edge lengths in {1,2}). We show that all nine of these questions are complete for the class Exists-R, defined by the Existential Theory of the Reals, or its complement Forall-R; in particular, each problem is (co)NP-hard.
One of these nine results - that realization of unit-distance graphs is Exists-R-complete - was shown previously by Schaefer (2013), but the other eight are new. We strengthen several prior results. Matchstick graph realization was known to be NP-hard (Eades & Wormald 1990, or Cabello et al. 2007), but its membership in NP remained open; we show it is complete for the (possibly) larger class Exists-R. Global rigidity of graphs with edge lengths in {1,2} was known to be coNP-hard (Saxe 1979); we show it is Forall-R-complete.
The majority of the paper is devoted to proving an analog of Kempe's Universality Theorem - informally, "there is a linkage to sign your name" - for globally noncrossing linkages. In particular, we show that any polynomial curve phi(x,y)=0 can be traced by a noncrossing linkage, settling an open problem from 2004. More generally, we show that the nontrivial regions in the plane that may be traced by a noncrossing linkage are precisely the compact semialgebraic regions. Thus, no drawing power is lost by restricting to noncrossing linkages. We prove analogous results for matchstick linkages and unit-distance linkages as well.

Zachary Abel, Erik D. Demaine, Martin L. Demaine, Sarah Eisenstat, Jayson Lynch, and Tao B. Schardl. Who Needs Crossings? Hardness of Plane Graph Rigidity. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{abel_et_al:LIPIcs.SoCG.2016.3, author = {Abel, Zachary and Demaine, Erik D. and Demaine, Martin L. and Eisenstat, Sarah and Lynch, Jayson and Schardl, Tao B.}, title = {{Who Needs Crossings? Hardness of Plane Graph Rigidity}}, booktitle = {32nd International Symposium on Computational Geometry (SoCG 2016)}, pages = {3:1--3:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-009-5}, ISSN = {1868-8969}, year = {2016}, volume = {51}, editor = {Fekete, S\'{a}ndor and Lubiw, Anna}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2016.3}, URN = {urn:nbn:de:0030-drops-58951}, doi = {10.4230/LIPIcs.SoCG.2016.3}, annote = {Keywords: Graph Drawing, Graph Rigidity Theory, Graph Global Rigidity, Linkages, Complexity Theory, Computational Geometry} }