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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 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Given a planar straight-line graph G=(V,E) in R^2, a circumscribing polygon of G is a simple polygon P whose vertex set is V, and every edge in E is either an edge or an internal diagonal of P. A circumscribing polygon is a polygonization for G if every edge in E is an edge of P.
We prove that every arrangement of n disjoint line segments in the plane has a subset of size Omega(sqrt{n}) that admits a circumscribing polygon, which is the first improvement on this bound in 20 years. We explore relations between circumscribing polygons and other problems in combinatorial geometry, and generalizations to R^3.
We show that it is NP-complete to decide whether a given graph G admits a circumscribing polygon, even if G is 2-regular. Settling a 30-year old conjecture by Rappaport, we also show that it is NP-complete to determine whether a geometric matching admits a polygonization.

Hugo A. Akitaya, Matias Korman, Mikhail Rudoy, Diane L. Souvaine, and Csaba D. Tóth. Circumscribing Polygons and Polygonizations for Disjoint Line Segments. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{akitaya_et_al:LIPIcs.SoCG.2019.9, author = {Akitaya, Hugo A. and Korman, Matias and Rudoy, Mikhail and Souvaine, Diane L. and T\'{o}th, Csaba D.}, title = {{Circumscribing Polygons and Polygonizations for Disjoint Line Segments}}, booktitle = {35th International Symposium on Computational Geometry (SoCG 2019)}, pages = {9:1--9:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-104-7}, ISSN = {1868-8969}, year = {2019}, volume = {129}, editor = {Barequet, Gill and Wang, Yusu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.9}, URN = {urn:nbn:de:0030-drops-104136}, doi = {10.4230/LIPIcs.SoCG.2019.9}, annote = {Keywords: circumscribing polygon, Hamiltonicity, extremal combinatorics} }

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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 analyze the computational complexity of optimally playing the two-player board game Push Fight, generalized to an arbitrary board and number of pieces. We prove that the game is PSPACE-hard to decide who will win from a given position, even for simple (almost rectangular) hole-free boards. We also analyze the mate-in-1 problem: can the player win in a single turn? One turn in Push Fight consists of up to two "moves" followed by a mandatory "push". With these rules, or generalizing the number of allowed moves to any constant, we show mate-in-1 can be solved in polynomial time. If, however, the number of moves per turn is part of the input, the problem becomes NP-complete. On the other hand, without any limit on the number of moves per turn, the problem becomes polynomially solvable again.

Jeffrey Bosboom, Erik D. Demaine, and Mikhail Rudoy. Computational Complexity of Generalized Push Fight. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 11:1-11:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bosboom_et_al:LIPIcs.FUN.2018.11, author = {Bosboom, Jeffrey and Demaine, Erik D. and Rudoy, Mikhail}, title = {{Computational Complexity of Generalized Push Fight}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {11:1--11:21}, 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.2018.11}, URN = {urn:nbn:de:0030-drops-88029}, doi = {10.4230/LIPIcs.FUN.2018.11}, annote = {Keywords: board games, hardness, mate-in-one} }

Document

**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 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

In this paper, we introduce a new problem called Tree-Residue Vertex-Breaking (TRVB): given a multigraph G some of whose vertices are marked "breakable," is it possible to convert G into a tree via a sequence of "vertex-breaking" operations (replacing a degree-k breakable vertex by k degree-1 vertices, disconnecting the k incident edges)?
We characterize the computational complexity of TRVB with any combination of the following additional constraints: G must be planar, G must be a simple graph, the degree of every breakable vertex must belong to an allowed list B, and the degree of every unbreakable vertex must belong to an allowed list U. The two results which we expect to be most generally applicable are that (1) TRVB is polynomially solvable when breakable vertices are restricted to have degree at most 3; and (2) for any k >= 4, TRVB is NP-complete when the given multigraph is restricted to be planar and to consist entirely of degree-k breakable vertices. To demonstrate the use of TRVB, we give a simple proof of the known result that Hamiltonicity in max-degree-3 square grid graphs is NP-hard.
We also demonstrate a connection between TRVB and the Hypergraph Spanning Tree problem. This connection allows us to show that the Hypergraph Spanning Tree problem in k-uniform 2-regular hypergraphs is NP-complete for any k >= 4, even when the incidence graph of the hypergraph is planar.

Erik D. Demaine and Mikhail Rudoy. Tree-Residue Vertex-Breaking: a new tool for proving hardness. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{demaine_et_al:LIPIcs.SWAT.2018.32, author = {Demaine, Erik D. and Rudoy, Mikhail}, title = {{Tree-Residue Vertex-Breaking: a new tool for proving hardness}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {32:1--32:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.32}, URN = {urn:nbn:de:0030-drops-88586}, doi = {10.4230/LIPIcs.SWAT.2018.32}, annote = {Keywords: NP-hardness, graphs, Hamiltonicity, hypergraph spanning tree} }

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**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

In this paper, we prove that optimally solving an n x n x n Rubik's Cube is NP-complete by reducing from the Hamiltonian Cycle problem in square grid graphs. This improves the previous result that optimally solving an n x n x n Rubik's Cube with missing stickers is NP-complete. We prove this result first for the simpler case of the Rubik's Square--an n x n x 1 generalization of the Rubik's Cube--and then proceed with a similar but more complicated proof for the Rubik's Cube case. Our results hold both when the goal is make the sides monochromatic and when the goal is to put each sticker into a specific location.

Erik D. Demaine, Sarah Eisenstat, and Mikhail Rudoy. Solving the Rubik's Cube Optimally is NP-complete. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 24:1-24:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{demaine_et_al:LIPIcs.STACS.2018.24, author = {Demaine, Erik D. and Eisenstat, Sarah and Rudoy, Mikhail}, title = {{Solving the Rubik's Cube Optimally is NP-complete}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {24:1--24:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.24}, URN = {urn:nbn:de:0030-drops-85336}, doi = {10.4230/LIPIcs.STACS.2018.24}, annote = {Keywords: combinatorial puzzles, NP-hardness, group theory, Hamiltonicity} }

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