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

In the k-Recoloring problem, we are given two (vertex-)colorings of a graph using k colors, and asked to transform one into the other by recoloring only one vertex at a time, while at all times maintaining a proper k-coloring. This problem is known to be solvable in polynomial time if k ≤ 3, and is PSPACE-complete if k ≥ 4. In this paper, we consider a (directed) recolorability constraint on the k colors, which forbids some pairs of colors to be recolored directly. The recolorability constraint is given in terms of a digraph R, whose vertices correspond to the colors and whose arcs represent the pairs of colors that can be recolored directly. We provide algorithms for the problem based on the structure of recolorability constraints R, showing that the problem is solvable in linear time when R is a directed cycle or is in a class of multitrees.

Soichiro Fujii, Yuni Iwamasa, Kei Kimura, and Akira Suzuki. Algorithms for Coloring Reconfiguration Under Recolorability Digraphs. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 4:1-4:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fujii_et_al:LIPIcs.ISAAC.2022.4, author = {Fujii, Soichiro and Iwamasa, Yuni and Kimura, Kei and Suzuki, Akira}, title = {{Algorithms for Coloring Reconfiguration Under Recolorability Digraphs}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {4:1--4:19}, 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.2022.4}, URN = {urn:nbn:de:0030-drops-172896}, doi = {10.4230/LIPIcs.ISAAC.2022.4}, annote = {Keywords: combinatorial reconfiguration, graph coloring, recolorability, recoloring} }

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

Various forms of sorting problems have been studied over the years. Recently, two kinds of sorting puzzle apps are popularized. In these puzzles, we are given a set of bins filled with colored units, balls or water, and some empty bins. These puzzles allow us to move colored units from a bin to another when the colors involved match in some way or the target bin is empty. The goal of these puzzles is to sort all the color units in order. We investigate computational complexities of these puzzles. We first show that these two puzzles are essentially the same from the viewpoint of solvability. That is, an instance is sortable by ball-moves if and only if it is sortable by water-moves. We also show that every yes-instance has a solution of polynomial length, which implies that these puzzles belong to NP . We then show that these puzzles are NP-complete. For some special cases, we give polynomial-time algorithms. We finally consider the number of empty bins sufficient for making all instances solvable and give non-trivial upper and lower bounds in terms of the number of filled bins and the capacity of bins.

Takehiro Ito, Jun Kawahara, Shin-ichi Minato, Yota Otachi, Toshiki Saitoh, Akira Suzuki, Ryuhei Uehara, Takeaki Uno, Katsuhisa Yamanaka, and Ryo Yoshinaka. Sorting Balls and Water: Equivalence and Computational Complexity. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 16:1-16:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ito_et_al:LIPIcs.FUN.2022.16, author = {Ito, Takehiro and Kawahara, Jun and Minato, Shin-ichi and Otachi, Yota and Saitoh, Toshiki and Suzuki, Akira and Uehara, Ryuhei and Uno, Takeaki and Yamanaka, Katsuhisa and Yoshinaka, Ryo}, title = {{Sorting Balls and Water: Equivalence and Computational Complexity}}, booktitle = {11th International Conference on Fun with Algorithms (FUN 2022)}, pages = {16:1--16:17}, 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.2022.16}, URN = {urn:nbn:de:0030-drops-159867}, doi = {10.4230/LIPIcs.FUN.2022.16}, annote = {Keywords: Ball sort puzzle, recreational mathematics, sorting pairs in bins, water sort puzzle} }

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**Published in:** LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

We investigate the complexity of finding a transformation from a given spanning tree in a graph to another given spanning tree in the same graph via a sequence of edge flips. The exchange property of the matroid bases immediately yields that such a transformation always exists if we have no constraints on spanning trees. In this paper, we wish to find a transformation which passes through only spanning trees satisfying some constraint. Our focus is bounding either the maximum degree or the diameter of spanning trees, and we give the following results. The problem with a lower bound on maximum degree is solvable in polynomial time, while the problem with an upper bound on maximum degree is PSPACE-complete. The problem with a lower bound on diameter is NP-hard, while the problem with an upper bound on diameter is solvable in polynomial time.

Nicolas Bousquet, Takehiro Ito, Yusuke Kobayashi, Haruka Mizuta, Paul Ouvrard, Akira Suzuki, and Kunihiro Wasa. Reconfiguration of Spanning Trees with Degree Constraint or Diameter Constraint. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 15:1-15:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{bousquet_et_al:LIPIcs.STACS.2022.15, author = {Bousquet, Nicolas and Ito, Takehiro and Kobayashi, Yusuke and Mizuta, Haruka and Ouvrard, Paul and Suzuki, Akira and Wasa, Kunihiro}, title = {{Reconfiguration of Spanning Trees with Degree Constraint or Diameter Constraint}}, booktitle = {39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)}, pages = {15:1--15:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-222-8}, ISSN = {1868-8969}, year = {2022}, volume = {219}, editor = {Berenbrink, Petra and Monmege, Benjamin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.15}, URN = {urn:nbn:de:0030-drops-158253}, doi = {10.4230/LIPIcs.STACS.2022.15}, annote = {Keywords: combinatorial reconfiguration, spanning trees, PSPACE, polynomial-time algorithms} }

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**Published in:** LIPIcs, Volume 180, 15th International Symposium on Parameterized and Exact Computation (IPEC 2020)

Non-deterministic constraint logic (NCL) is a simple model of computation based on orientations of a constraint graph with edge weights and vertex demands. NCL captures PSPACE and has been a useful tool for proving algorithmic hardness of many puzzles, games, and reconfiguration problems. In particular, its usefulness stems from the fact that it remains PSPACE-complete even under severe restrictions of the weights (e.g., only edge-weights one and two are needed) and the structure of the constraint graph (e.g., planar AND/OR graphs of bounded bandwidth). While such restrictions on the structure of constraint graphs do not seem to limit the expressiveness of NCL, the building blocks of the constraint graphs cannot be limited without losing expressiveness: We consider as parameters the number of weight-one edges and the number of weight-two edges of a constraint graph, as well as the number of AND or OR vertices of an AND/OR constraint graph. We show that NCL is fixed-parameter tractable (FPT) for any of these parameters. In particular, for NCL parameterized by the number of weight-one edges or the number of AND vertices, we obtain a linear kernel. It follows that, in a sense, NCL as introduced by Hearn and Demaine is defined in the most economical way for the purpose of capturing PSPACE.

Tatsuhiko Hatanaka, Felix Hommelsheim, Takehiro Ito, Yusuke Kobayashi, Moritz Mühlenthaler, and Akira Suzuki. Fixed-Parameter Algorithms for Graph Constraint Logic. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 15:1-15:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hatanaka_et_al:LIPIcs.IPEC.2020.15, author = {Hatanaka, Tatsuhiko and Hommelsheim, Felix and Ito, Takehiro and Kobayashi, Yusuke and M\"{u}hlenthaler, Moritz and Suzuki, Akira}, title = {{Fixed-Parameter Algorithms for Graph Constraint Logic}}, booktitle = {15th International Symposium on Parameterized and Exact Computation (IPEC 2020)}, pages = {15:1--15:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-172-6}, ISSN = {1868-8969}, year = {2020}, volume = {180}, editor = {Cao, Yixin and Pilipczuk, Marcin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2020.15}, URN = {urn:nbn:de:0030-drops-133182}, doi = {10.4230/LIPIcs.IPEC.2020.15}, annote = {Keywords: Combinatorial Reconfiguration, Nondeterministic Constraint Logic, Fixed Parameter Tractability} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Let G be a graph and T₁,T₂ be two spanning trees of G. We say that T₁ can be transformed into T₂ via an edge flip if there exist two edges e ∈ T₁ and f in T₂ such that T₂ = (T₁⧵e) ∪ f. Since spanning trees form a matroid, one can indeed transform a spanning tree into any other via a sequence of edge flips, as observed in [Takehiro Ito et al., 2011].
We investigate the problem of determining, given two spanning trees T₁,T₂ with an additional property Π, if there exists an edge flip transformation from T₁ to T₂ keeping property Π all along.
First we show that determining if there exists a transformation from T₁ to T₂ such that all the trees of the sequence have at most k (for any fixed k ≥ 3) leaves is PSPACE-complete.
We then prove that determining if there exists a transformation from T₁ to T₂ such that all the trees of the sequence have at least k leaves (where k is part of the input) is PSPACE-complete even restricted to split, bipartite or planar graphs. We complete this result by showing that the problem becomes polynomial for cographs, interval graphs and when k = n-2.

Nicolas Bousquet, Takehiro Ito, Yusuke Kobayashi, Haruka Mizuta, Paul Ouvrard, Akira Suzuki, and Kunihiro Wasa. Reconfiguration of Spanning Trees with Many or Few Leaves. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 24:1-24:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bousquet_et_al:LIPIcs.ESA.2020.24, author = {Bousquet, Nicolas and Ito, Takehiro and Kobayashi, Yusuke and Mizuta, Haruka and Ouvrard, Paul and Suzuki, Akira and Wasa, Kunihiro}, title = {{Reconfiguration of Spanning Trees with Many or Few Leaves}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {24:1--24:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.24}, URN = {urn:nbn:de:0030-drops-128909}, doi = {10.4230/LIPIcs.ESA.2020.24}, annote = {Keywords: combinatorial reconfiguration, spanning trees, PSPACE, polynomial-time algorithms} }

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**Published in:** LIPIcs, Volume 154, 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)

A k-coloring of a graph maps each vertex of the graph to a color in {1, 2, …, k}, such that no two adjacent vertices receive the same color. Given a k-coloring of a graph, a Kempe change produces a new k-coloring by swapping the colors in a bicolored connected component. We investigate the complexity of finding the smallest number of Kempe changes needed to transform a given k-coloring into another given k-coloring. We show that this problem admits a polynomial-time dynamic programming algorithm on path graphs, which turns out to be highly non-trivial. Furthermore, the problem is NP-hard even on star graphs and we show that on such graphs it admits a constant-factor approximation algorithm and is fixed-parameter tractable when parameterized by the number k of colors. The hardness result as well as the algorithmic results are based on the notion of a canonical transformation.

Marthe Bonamy, Marc Heinrich, Takehiro Ito, Yusuke Kobayashi, Haruka Mizuta, Moritz Mühlenthaler, Akira Suzuki, and Kunihiro Wasa. Shortest Reconfiguration of Colorings Under Kempe Changes. In 37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 154, pp. 35:1-35:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bonamy_et_al:LIPIcs.STACS.2020.35, author = {Bonamy, Marthe and Heinrich, Marc and Ito, Takehiro and Kobayashi, Yusuke and Mizuta, Haruka and M\"{u}hlenthaler, Moritz and Suzuki, Akira and Wasa, Kunihiro}, title = {{Shortest Reconfiguration of Colorings Under Kempe Changes}}, booktitle = {37th International Symposium on Theoretical Aspects of Computer Science (STACS 2020)}, pages = {35:1--35:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-140-5}, ISSN = {1868-8969}, year = {2020}, volume = {154}, editor = {Paul, Christophe and Bl\"{a}ser, Markus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2020.35}, URN = {urn:nbn:de:0030-drops-118961}, doi = {10.4230/LIPIcs.STACS.2020.35}, annote = {Keywords: Combinatorial Reconfiguration, Graph Algorithms, Graph Coloring, Kempe Equivalence} }

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**Published in:** LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Coloring reconfiguration is one of the most well-studied reconfiguration problems. In the problem, we are given two (vertex-)colorings of a graph using at most k colors, and asked to determine whether there exists a transformation between them by recoloring only a single vertex at a time, while maintaining a k-coloring throughout. It is known that this problem is solvable in linear time for any graph if k <=3, while is PSPACE-complete for a fixed k >= 4. In this paper, we further investigate the problem from the viewpoint of recolorability constraints, which forbid some pairs of colors to be recolored directly. More specifically, the recolorability constraint is given in terms of an undirected graph R such that each node in R corresponds to a color, and each edge in R represents a pair of colors that can be recolored directly. In this paper, we give a linear-time algorithm to solve the problem under such a recolorability constraint if R is of maximum degree at most two. In addition, we show that the minimum number of recoloring steps required for a desired transformation can be computed in linear time for a yes-instance. We note that our results generalize the known positive ones for coloring reconfiguration.

Hiroki Osawa, Akira Suzuki, Takehiro Ito, and Xiao Zhou. Algorithms for Coloring Reconfiguration Under Recolorability Constraints. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 37:1-37:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{osawa_et_al:LIPIcs.ISAAC.2018.37, author = {Osawa, Hiroki and Suzuki, Akira and Ito, Takehiro and Zhou, Xiao}, title = {{Algorithms for Coloring Reconfiguration Under Recolorability Constraints}}, booktitle = {29th International Symposium on Algorithms and Computation (ISAAC 2018)}, pages = {37:1--37:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-094-1}, ISSN = {1868-8969}, year = {2018}, volume = {123}, editor = {Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.37}, URN = {urn:nbn:de:0030-drops-99850}, doi = {10.4230/LIPIcs.ISAAC.2018.37}, annote = {Keywords: combinatorial reconfiguration, graph algorithm, graph coloring} }

Document

**Published in:** LIPIcs, Volume 92, 28th International Symposium on Algorithms and Computation (ISAAC 2017)

For an integer k \ge 1, k-coloring reconfiguration is one of the most well-studied reconfiguration problems, defined as follows: In the problem, we are given two (vertex-)colorings of a graph using k colors, and asked to transform one into the other by recoloring only one vertex at a time, while at all times maintaining a proper coloring. The problem is known to be PSPACE-complete if k \ge 4, and solvable for any graph in polynomial time if k \le 3. In this paper, we introduce a recolorability constraint on the k colors, which forbids some pairs of colors to be recolored directly. The recolorability constraint is given in terms of an undirected graph R such that each node in R corresponds to a color and each edge in R represents a pair of colors that can be recolored directly. We study the hardness of the problem based on the structure of recolorability constraints R. More specifically, we prove that the problem is PSPACE-complete if R is of maximum degree at least four, or has a connected component containing more than one cycle.

Hiroki Osawa, Akira Suzuki, Takehiro Ito, and Xiao Zhou. Complexity of Coloring Reconfiguration under Recolorability Constraints. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 62:1-62:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{osawa_et_al:LIPIcs.ISAAC.2017.62, author = {Osawa, Hiroki and Suzuki, Akira and Ito, Takehiro and Zhou, Xiao}, title = {{Complexity of Coloring Reconfiguration under Recolorability Constraints}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {62:1--62:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-054-5}, ISSN = {1868-8969}, year = {2017}, volume = {92}, editor = {Okamoto, Yoshio and Tokuyama, Takeshi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2017.62}, URN = {urn:nbn:de:0030-drops-82588}, doi = {10.4230/LIPIcs.ISAAC.2017.62}, annote = {Keywords: combinatorial reconfiguration, graph coloring, PSPACE-complete} }

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