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Complete Volume

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

LIPIcs, Volume 100, FUN'18, Complete Volume

Hiro Ito, Stefano Leonardi, Linda Pagli, and Giuseppe Prencipe. LIPIcs, Volume 100, FUN'18, Complete Volume. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@Proceedings{ito_et_al:LIPIcs.FUN.2018, title = {{LIPIcs, Volume 100, FUN'18, Complete Volume}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, 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}, URN = {urn:nbn:de:0030-drops-89311}, doi = {10.4230/LIPIcs.FUN.2018}, annote = {Keywords: Theory of computation, Complexity classes, Algorithm design techniques, Computability, Approximation algorithms analysis, Mathematics of computing} }

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Front Matter

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

Front Matter, Table of Contents, Preface, Conference Organization

Hiro Ito, Stefano Leonardi, Linda Pagli, and Giuseppe Prencipe. Front Matter, Table of Contents, Preface, Conference Organization. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 0:i-0:xi, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ito_et_al:LIPIcs.FUN.2018.0, author = {Ito, Hiro and Leonardi, Stefano and Pagli, Linda and Prencipe, Giuseppe}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {9th International Conference on Fun with Algorithms (FUN 2018)}, pages = {0:i--0:xi}, 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.2018.0}, URN = {urn:nbn:de:0030-drops-87914}, doi = {10.4230/LIPIcs.FUN.2018.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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**Published in:** LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

In this paper, we introduce a natural class of multigraphs called hierarchical-scale-free (HSF) multigraphs, and consider constant-time testability on the class. We show that a very wide subclass of HSF is hyperfinite. Based on this result, an algorithm for a deterministic partitioning oracle can be constructed. We conclude by showing that every property is constant-time testable on the above subclass of HSF. This algorithm utilizes findings by Newman and Sohler of STOC'11. However, their algorithm is based on a bounded-degree model, while it is known that actual scale-free networks usually include hubs, which have a very large degree. HSF is based on scale-free properties and includes such hubs. This is the first universal result of constant-time testability on a class of graphs made by a model of scale-free networks, and it has the potential to be applicable on a very wide range of scale-free networks.

Hiro Ito. Every Property Is Testable on a Natural Class of Scale-Free Multigraphs. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 51:1-51:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ito:LIPIcs.ESA.2016.51, author = {Ito, Hiro}, title = {{Every Property Is Testable on a Natural Class of Scale-Free Multigraphs}}, booktitle = {24th Annual European Symposium on Algorithms (ESA 2016)}, pages = {51:1--51:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-015-6}, ISSN = {1868-8969}, year = {2016}, volume = {57}, editor = {Sankowski, Piotr and Zaroliagis, Christos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.51}, URN = {urn:nbn:de:0030-drops-63937}, doi = {10.4230/LIPIcs.ESA.2016.51}, annote = {Keywords: constant-time algorithms, scale-free networks, complex networks, isolated cliques, hyperfinite} }

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**Published in:** LIPIcs, Volume 49, 8th International Conference on Fun with Algorithms (FUN 2016)

The cake-cutting problem refers to the issue of dividing a cake into pieces and distributing them to players who have different value measures related to the cake, and who feel that their portions should be "fair." The fairness criterion specifies that in situations where n is the number of players, each player should receive his/her portion with at least 1/n of the cake value in his/her measure. In this paper, we show algorithms for solving the cake-cutting problem in sublinear-time. More specifically, we preassign fair portions to o(n) players in o(n)-time, and minimize the damage to the rest of the players. All currently known algorithms require Omega(n)-time, even when assigning a portion to just one player, and it is nontrivial to revise these algorithms to run in o(n)-time since many of the remaining players, who have not been asked any queries, may not be satisfied with the remaining cake. To challenge this problem, we begin by providing a framework for solving the cake-cutting problem in sublinear-time. Generally speaking, solving a problem in sublinear-time requires the use of approximations. However, in our framework, we introduce the concept of "epsilon n-victims," which means that (epsilon x n) players (victims) may not get fair portions, where 0< epsilon =< 1 is an arbitrary constant. In our framework, an algorithm consists of the following two parts: In the first (Preassigning) part, it distributes fair portions to r < n players in o(n)-time. In the second (Completion) part, it distributes fair portions to the remaining n-r players except for the (epsilon x n) victims in poly(n)-time. There are two variations on the r players in the first part. Specifically, whether they can or cannot be designated. We will then present algorithms in this framework. In particular, an O(r/epsilon)-time algorithm for r =< (epsilon x n)/127 undesignated players with (epsilon x n)-victims, and an tilde{O}(r^2/epsilon)-time algorithm for r =< (epsilon x e^(((sqrt(ln(n)))/7) designated players and epsilon =< 1/e with (epsilon x n)-victims are presented.

Hiro Ito and Takahiro Ueda. How to Solve the Cake-Cutting Problem in Sublinear Time. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 21:1-21:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ito_et_al:LIPIcs.FUN.2016.21, author = {Ito, Hiro and Ueda, Takahiro}, title = {{How to Solve the Cake-Cutting Problem in Sublinear Time}}, booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)}, pages = {21:1--21:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-005-7}, ISSN = {1868-8969}, year = {2016}, volume = {49}, editor = {Demaine, Erik D. and Grandoni, Fabrizio}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2016.21}, URN = {urn:nbn:de:0030-drops-58639}, doi = {10.4230/LIPIcs.FUN.2016.21}, annote = {Keywords: sublinear-time algorithms, cake-cutting problem, simple fair, preassign, approximation} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

In this paper, we consider a transformation of k disjoint paths in a graph. For a graph and a pair of k disjoint paths 𝒫 and 𝒬 connecting the same set of terminal pairs, we aim to determine whether 𝒫 can be transformed to 𝒬 by repeatedly replacing one path with another path so that the intermediates are also k disjoint paths. The problem is called Disjoint Paths Reconfiguration. We first show that Disjoint Paths Reconfiguration is PSPACE-complete even when k = 2. On the other hand, we prove that, when the graph is embedded on a plane and all paths in 𝒫 and 𝒬 connect the boundaries of two faces, Disjoint Paths Reconfiguration can be solved in polynomial time. The algorithm is based on a topological characterization for rerouting curves on a plane using the algebraic intersection number. We also consider a transformation of disjoint s-t paths as a variant. We show that the disjoint s-t paths reconfiguration problem in planar graphs can be determined in polynomial time, while the problem is PSPACE-complete in general.

Takehiro Ito, Yuni Iwamasa, Naonori Kakimura, Yusuke Kobayashi, Shun-ichi Maezawa, Yuta Nozaki, Yoshio Okamoto, and Kenta Ozeki. Rerouting Planar Curves and Disjoint Paths. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 81:1-81:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ito_et_al:LIPIcs.ICALP.2023.81, author = {Ito, Takehiro and Iwamasa, Yuni and Kakimura, Naonori and Kobayashi, Yusuke and Maezawa, Shun-ichi and Nozaki, Yuta and Okamoto, Yoshio and Ozeki, Kenta}, title = {{Rerouting Planar Curves and Disjoint Paths}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {81:1--81:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.81}, URN = {urn:nbn:de:0030-drops-181339}, doi = {10.4230/LIPIcs.ICALP.2023.81}, annote = {Keywords: Disjoint paths, combinatorial reconfiguration, planar graphs} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

We prove that the computation of a combinatorial shortest path between two vertices of a graph associahedron, introduced by Carr and Devadoss, is NP-hard. This resolves an open problem raised by Cardinal. A graph associahedron is a generalization of the well-known associahedron. The associahedron is obtained as the graph associahedron of a path. It is a tantalizing and important open problem in theoretical computer science whether the computation of a combinatorial shortest path between two vertices of the associahedron can be done in polynomial time, which is identical to the computation of the flip distance between two triangulations of a convex polygon, and the rotation distance between two rooted binary trees. Our result shows that a certain generalized approach to tackling this open problem is not promising. As a corollary of our theorem, we prove that the computation of a combinatorial shortest path between two vertices of a polymatroid base polytope cannot be done in polynomial time unless P = NP. Since a combinatorial shortest path on the matroid base polytope can be computed in polynomial time, our result reveals an unexpected contrast between matroids and polymatroids.

Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, Shun-ichi Maezawa, Yuta Nozaki, and Yoshio Okamoto. Hardness of Finding Combinatorial Shortest Paths on Graph Associahedra. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 82:1-82:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ito_et_al:LIPIcs.ICALP.2023.82, author = {Ito, Takehiro and Kakimura, Naonori and Kamiyama, Naoyuki and Kobayashi, Yusuke and Maezawa, Shun-ichi and Nozaki, Yuta and Okamoto, Yoshio}, title = {{Hardness of Finding Combinatorial Shortest Paths on Graph Associahedra}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {82:1--82:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.82}, URN = {urn:nbn:de:0030-drops-181344}, doi = {10.4230/LIPIcs.ICALP.2023.82}, annote = {Keywords: Graph associahedra, combinatorial shortest path, NP-hardness, polymatroids} }

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**Published in:** LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

In 1973, Fisk proved that any 4-coloring of a 3-colorable triangulation of the 2-sphere can be obtained from any 3-coloring by a sequence of Kempe-changes. On the other hand, in the case where we are only allowed to recolor a single vertex in each step, which is a special case of a Kempe-change, there exists a 4-coloring that cannot be obtained from any 3-coloring.
In this paper, we present a linear-time checkable characterization of a 4-coloring of a 3-colorable triangulation of the 2-sphere that can be obtained from a 3-coloring by a sequence of recoloring operations at single vertices. In addition, we develop a quadratic-time algorithm to find such a recoloring sequence if it exists; our proof implies that we can always obtain a quadratic length recoloring sequence. We also present a linear-time checkable criterion for a 3-colorable triangulation of the 2-sphere that all 4-colorings can be obtained from a 3-coloring by such a sequence. Moreover, we consider a high-dimensional setting. As a natural generalization of our first result, we obtain a polynomial-time checkable characterization of a k-coloring of a (k-1)-colorable triangulation of the (k-2)-sphere that can be obtained from a (k-1)-coloring by a sequence of recoloring operations at single vertices and the corresponding algorithmic result. Furthermore, we show that the problem of deciding whether, for given two (k+1)-colorings of a (k-1)-colorable triangulation of the (k-2)-sphere, one can be obtained from the other by such a sequence is PSPACE-complete for any fixed k ≥ 4. Our results above can be rephrased as new results on the computational problems named k-Recoloring and Connectedness of k-Coloring Reconfiguration Graph, which are fundamental problems in the field of combinatorial reconfiguration.

Takehiro Ito, Yuni Iwamasa, Yusuke Kobayashi, Shun-ichi Maezawa, Yuta Nozaki, Yoshio Okamoto, and Kenta Ozeki. Reconfiguration of Colorings in Triangulations of the Sphere. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 43:1-43:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{ito_et_al:LIPIcs.SoCG.2023.43, author = {Ito, Takehiro and Iwamasa, Yuni and Kobayashi, Yusuke and Maezawa, Shun-ichi and Nozaki, Yuta and Okamoto, Yoshio and Ozeki, Kenta}, title = {{Reconfiguration of Colorings in Triangulations of the Sphere}}, booktitle = {39th International Symposium on Computational Geometry (SoCG 2023)}, pages = {43:1--43:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-273-0}, ISSN = {1868-8969}, year = {2023}, volume = {258}, editor = {Chambers, Erin W. and Gudmundsson, Joachim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.43}, URN = {urn:nbn:de:0030-drops-178936}, doi = {10.4230/LIPIcs.SoCG.2023.43}, annote = {Keywords: Graph coloring, Triangulation of the sphere, Combinatorial reconfiguration} }

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

Given a graph and two vertex sets satisfying a certain feasibility condition, a reconfiguration problem asks whether we can reach one vertex set from the other by repeating prescribed modification steps while maintaining feasibility. In this setting, Mouawad et al. [IPEC 2014] presented an algorithmic meta-theorem for reconfiguration problems that says if the feasibility can be expressed in monadic second-order logic (MSO), then the problem is fixed-parameter tractable parameterized by treewidth + 𝓁, where 𝓁 is the number of steps allowed to reach the target set. On the other hand, it is shown by Wrochna [J. Comput. Syst. Sci. 2018] that if 𝓁 is not part of the parameter, then the problem is PSPACE-complete even on graphs of bounded bandwidth.
In this paper, we present the first algorithmic meta-theorems for the case where 𝓁 is not part of the parameter, using some structural graph parameters incomparable with bandwidth. We show that if the feasibility is defined in MSO, then the reconfiguration problem under the so-called token jumping rule is fixed-parameter tractable parameterized by neighborhood diversity. We also show that the problem is fixed-parameter tractable parameterized by treedepth + k, where k is the size of sets being transformed. We finally complement the positive result for treedepth by showing that the problem is PSPACE-complete on forests of depth 3.

Tatsuya Gima, Takehiro Ito, Yasuaki Kobayashi, and Yota Otachi. Algorithmic Meta-Theorems for Combinatorial Reconfiguration Revisited. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 61:1-61:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gima_et_al:LIPIcs.ESA.2022.61, author = {Gima, Tatsuya and Ito, Takehiro and Kobayashi, Yasuaki and Otachi, Yota}, title = {{Algorithmic Meta-Theorems for Combinatorial Reconfiguration Revisited}}, booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)}, pages = {61:1--61:15}, 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.2022.61}, URN = {urn:nbn:de:0030-drops-169991}, doi = {10.4230/LIPIcs.ESA.2022.61}, annote = {Keywords: Combinatorial reconfiguration, monadic second-order logic, fixed-parameter tractability, treedepth, neighborhood diversity} }

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**Published in:** LIPIcs, Volume 241, 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)

Directed Token Sliding asks, given a directed graph and two sets of pairwise nonadjacent vertices, whether one can reach from one set to the other by repeatedly applying a local operation that exchanges a vertex in the current set with one of its out-neighbors, while keeping the nonadjacency. It can be seen as a reconfiguration process where a token is placed on each vertex in the current set, and the local operation slides a token along an arc respecting its direction. Previously, such a problem was extensively studied on undirected graphs, where the edges have no directions and thus the local operation is symmetric. Directed Token Sliding is a generalization of its undirected variant since an undirected edge can be simulated by two arcs of opposite directions.
In this paper, we initiate the algorithmic study of Directed Token Sliding. We first observe that the problem is PSPACE-complete even if we forbid parallel arcs in opposite directions and that the problem on directed acyclic graphs is NP-complete and W[1]-hard parameterized by the size of the sets in consideration. We then show our main result: a linear-time algorithm for the problem on directed graphs whose underlying undirected graphs are trees, which are called polytrees. Such a result is also known for the undirected variant of the problem on trees [Demaine et al. TCS 2015], but the techniques used here are quite different because of the asymmetric nature of the directed problem. We present a characterization of yes-instances based on the existence of a certain set of directed paths, and then derive simple equivalent conditions from it by some observations, which yield an efficient algorithm. For the polytree case, we also present a quadratic-time algorithm that outputs, if the input is a yes-instance, one of the shortest reconfiguration sequences.

Takehiro Ito, Yuni Iwamasa, Yasuaki Kobayashi, Yu Nakahata, Yota Otachi, Masahiro Takahashi, and Kunihiro Wasa. Independent Set Reconfiguration on Directed Graphs. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 58:1-58:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ito_et_al:LIPIcs.MFCS.2022.58, author = {Ito, Takehiro and Iwamasa, Yuni and Kobayashi, Yasuaki and Nakahata, Yu and Otachi, Yota and Takahashi, Masahiro and Wasa, Kunihiro}, title = {{Independent Set Reconfiguration on Directed Graphs}}, booktitle = {47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022)}, pages = {58:1--58:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-256-3}, ISSN = {1868-8969}, year = {2022}, volume = {241}, editor = {Szeider, Stefan and Ganian, Robert and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2022.58}, URN = {urn:nbn:de:0030-drops-168567}, doi = {10.4230/LIPIcs.MFCS.2022.58}, annote = {Keywords: Combinatorial reconfiguration, token sliding, directed graph, independent set, graph algorithm} }

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Invited Talk

**Published in:** LIPIcs, Volume 223, 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)

Combinatorial reconfiguration studies reachability and related questions over the solution space formed by feasible solutions of an instance of a combinatorial search problem. For example, as the solution space for the satisfiability problem, we may consider the subgraph of the hypercube induced by the satisfying truth assignments of a given CNF formula. Then, the reachability problem for satisfiability is the problem of asking whether two given satisfying truth assignments are contained in the same connected component of the solution space. The study of reconfiguration problems has motivation from a variety of fields such as puzzles, statistical physics, and industry. In this decade, reconfiguration problems have been studied intensively for many central combinatorial search problems, such as satisfiability, independent set and coloring, from the algorithmic viewpoints. Many reconfiguration problems are PSPACE-complete in general, although several efficiently solvable cases have been obtained. In this talk, I will give a broad introduction of combinatorial reconfiguration.

Takehiro Ito. Invitation to Combinatorial Reconfiguration (Invited Talk). In 33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 223, p. 1:1, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ito:LIPIcs.CPM.2022.1, author = {Ito, Takehiro}, title = {{Invitation to Combinatorial Reconfiguration}}, booktitle = {33rd Annual Symposium on Combinatorial Pattern Matching (CPM 2022)}, pages = {1:1--1:1}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-234-1}, ISSN = {1868-8969}, year = {2022}, volume = {223}, editor = {Bannai, Hideo and Holub, Jan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CPM.2022.1}, URN = {urn:nbn:de:0030-drops-161281}, doi = {10.4230/LIPIcs.CPM.2022.1}, annote = {Keywords: Combinatorial reconfiguration, graph algorithm} }

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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 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Let Π₁, Π₂, …, Π_c be graph properties for a fixed integer c. Then, (Π₁, Π₂, …, Π_c)-Partition is the problem of asking whether the vertex set of a given graph can be partitioned into c subsets V₁, V₂, …, V_c such that the subgraph induced by V_i satisfies the graph property Π_i for every i ∈ {1,2, …, c}. Minimization and parameterized variants of (Π₁, Π₂, …, Π_c)-Partition have been studied for several specific graph properties, where the size of the vertex subset V₁ satisfying Π₁ is minimized or taken as a parameter. In this paper, we first show that the minimization variant is hard to approximate for any nontrivial additive hereditary graph properties, unless c = 2 and both Π₁ and Π₂ are classes of edgeless graphs. We then give FPT algorithms for the parameterized variant when restricted to the case where c = 2, Π₁ is a hereditary graph property, and Π₂ is the class of acyclic graphs.

Yuma Tamura, Takehiro Ito, and Xiao Zhou. Minimization and Parameterized Variants of Vertex Partition Problems on Graphs. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 40:1-40:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{tamura_et_al:LIPIcs.ISAAC.2020.40, author = {Tamura, Yuma and Ito, Takehiro and Zhou, Xiao}, title = {{Minimization and Parameterized Variants of Vertex Partition Problems on Graphs}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {40:1--40:13}, 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.2020.40}, URN = {urn:nbn:de:0030-drops-133844}, doi = {10.4230/LIPIcs.ISAAC.2020.40}, annote = {Keywords: Graph Algorithms, Approximability, Fixed-Parameter Tractability, Vertex Partition Problem, Feedback Vertex Set Problem} }

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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 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in polynomial time when the graph is outerplanar.

Takehiro Ito, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, and Yoshio Okamoto. Shortest Reconfiguration of Perfect Matchings via Alternating Cycles. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 61:1-61:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ito_et_al:LIPIcs.ESA.2019.61, author = {Ito, Takehiro and Kakimura, Naonori and Kamiyama, Naoyuki and Kobayashi, Yusuke and Okamoto, Yoshio}, title = {{Shortest Reconfiguration of Perfect Matchings via Alternating Cycles}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {61:1--61:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola 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.2019.61}, URN = {urn:nbn:de:0030-drops-111823}, doi = {10.4230/LIPIcs.ESA.2019.61}, annote = {Keywords: Matching, Combinatorial reconfiguration, Alternating cycles, Combinatorial shortest paths} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

In this paper, we study the problem of deciding if there is a transformation between two given minimum Steiner trees of an unweighted graph such that each transformation step respects a prescribed reconfiguration rule and results in another minimum Steiner tree of the graph. We consider two reconfiguration rules, both of which exchange a single vertex at a time, and generalize the known reconfiguration problem for shortest paths in an unweighted graph. This generalization implies that our problems under both reconfiguration rules are PSPACE-complete for bipartite graphs. We thus study the problems with respect to graph classes, and give some boundaries between the polynomial-time solvable and PSPACE-complete cases.

Haruka Mizuta, Tatsuhiko Hatanaka, Takehiro Ito, and Xiao Zhou. Reconfiguration of Minimum Steiner Trees via Vertex Exchanges. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 79:1-79:11, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{mizuta_et_al:LIPIcs.MFCS.2019.79, author = {Mizuta, Haruka and Hatanaka, Tatsuhiko and Ito, Takehiro and Zhou, Xiao}, title = {{Reconfiguration of Minimum Steiner Trees via Vertex Exchanges}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {79:1--79:11}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.79}, URN = {urn:nbn:de:0030-drops-110234}, doi = {10.4230/LIPIcs.MFCS.2019.79}, annote = {Keywords: Combinatorial reconfiguration, Graph algorithms, Steiner tree} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

We study the perfect matching reconfiguration problem: Given two perfect matchings of a graph, is there a sequence of flip operations that transforms one into the other? Here, a flip operation exchanges the edges in an alternating cycle of length four. We are interested in the complexity of this decision problem from the viewpoint of graph classes. We first prove that the problem is PSPACE-complete even for split graphs and for bipartite graphs of bounded bandwidth with maximum degree five. We then investigate polynomial-time solvable cases. Specifically, we prove that the problem is solvable in polynomial time for strongly orderable graphs (that include interval graphs and strongly chordal graphs), for outerplanar graphs, and for cographs (also known as P_4-free graphs). Furthermore, for each yes-instance from these graph classes, we show that a linear number of flip operations is sufficient and we can exhibit a corresponding sequence of flip operations in polynomial time.

Marthe Bonamy, Nicolas Bousquet, Marc Heinrich, Takehiro Ito, Yusuke Kobayashi, Arnaud Mary, Moritz Mühlenthaler, and Kunihiro Wasa. The Perfect Matching Reconfiguration Problem. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 80:1-80:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bonamy_et_al:LIPIcs.MFCS.2019.80, author = {Bonamy, Marthe and Bousquet, Nicolas and Heinrich, Marc and Ito, Takehiro and Kobayashi, Yusuke and Mary, Arnaud and M\"{u}hlenthaler, Moritz and Wasa, Kunihiro}, title = {{The Perfect Matching Reconfiguration Problem}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {80:1--80:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.80}, URN = {urn:nbn:de:0030-drops-110248}, doi = {10.4230/LIPIcs.MFCS.2019.80}, annote = {Keywords: Combinatorial Reconfiguration, Graph Algorithms, Perfect Matching} }

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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} }

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**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

A set of vertices in a graph is c-colorable if the subgraph induced by the set has a proper c-coloring. In this paper, we study the problem of finding a step-by-step transformation (reconfiguration) between two c-colorable sets in the same graph. This problem generalizes the well-studied Independent Set Reconfiguration problem. As the first step toward a systematic understanding of the complexity of this general problem, we study the problem on classes of perfect graphs. We first focus on interval graphs and give a combinatorial characterization of the distance between two c-colorable sets. This gives a linear-time algorithm for finding an actual shortest reconfiguration sequence for interval graphs. Since interval graphs are exactly the graphs that are simultaneously chordal and co-comparability, we then complement the positive result by showing that even deciding reachability is PSPACE-complete for chordal graphs and for co-comparability graphs. The hardness for chordal graphs holds even for split graphs. We also consider the case where c is a fixed constant and show that in such a case the reachability problem is polynomial-time solvable for split graphs but still PSPACE-complete for co-comparability graphs. The complexity of this case for chordal graphs remains unsettled. As by-products, our positive results give the first polynomial-time solvable cases (split graphs and interval graphs) for Feedback Vertex Set Reconfiguration.

Takehiro Ito and Yota Otachi. Reconfiguration of Colorable Sets in Classes of Perfect Graphs. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 27:1-27:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ito_et_al:LIPIcs.SWAT.2018.27, author = {Ito, Takehiro and Otachi, Yota}, title = {{Reconfiguration of Colorable Sets in Classes of Perfect Graphs}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {27:1--27:13}, 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.2018.27}, URN = {urn:nbn:de:0030-drops-88539}, doi = {10.4230/LIPIcs.SWAT.2018.27}, annote = {Keywords: reconfiguration, colorable set, perfect graph} }

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

The multi-service center problem is a variant of facility location problems. In the problem, we consider locating p facilities on a graph, each of which provides distinct service required by all vertices. Each vertex incurs the cost determined by the sum of the weighted distances to the p facilities. The aim of the problem is to minimize the maximum cost among all vertices. This problem is known to be NP-hard for general graphs, while it is solvable in polynomial time when p is a fixed constant. In this paper, we give sharp analyses for the complexity of the problem from the viewpoint of graph classes and weights on vertices. We first propose a polynomial-time algorithm for trees when p is a part of input. In contrast, we prove that the problem becomes strongly NP-hard even for cycles. We also show that when vertices are allowed to have negative weights, the problem becomes NP-hard for paths of only three vertices and strongly NP-hard for stars.

Takehiro Ito, Naonori Kakimura, and Yusuke Kobayashi. Complexity of the Multi-Service Center Problem. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 48:1-48:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ito_et_al:LIPIcs.ISAAC.2017.48, author = {Ito, Takehiro and Kakimura, Naonori and Kobayashi, Yusuke}, title = {{Complexity of the Multi-Service Center Problem}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {48:1--48:12}, 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.2017.48}, URN = {urn:nbn:de:0030-drops-82536}, doi = {10.4230/LIPIcs.ISAAC.2017.48}, annote = {Keywords: facility location, graph algorithm, multi-service location} }

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**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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**Published in:** LIPIcs, Volume 83, 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)

Let G be a graph such that each vertex has its list of available colors, and assume that each list is a subset of the common set consisting of k colors. For two given list colorings of G, we study the problem of transforming one into the other by changing only one vertex color assignment at a time, while at all times maintaining a list coloring. This problem is known to be PSPACE-complete even for bounded bandwidth graphs and a fixed constant k. In this paper, we study the fixed-parameter tractability of the problem when parameterized by several graph parameters. We first give a fixed-parameter algorithm for the problem when parameterized by k and the modular-width of an input graph. We next give a fixed-parameter algorithm for the shortest variant which computes the length of a shortest transformation when parameterized by k and the size of a minimum vertex cover of an input graph. As corollaries, we show that the problem for cographs and the shortest variant for split graphs are fixed-parameter tractable even when only k is taken as a parameter. On the other hand, we prove that the problem is W[1]-hard when parameterized only by the size of a minimum vertex cover of an input graph.

Tatsuhiko Hatanaka, Takehiro Ito, and Xiao Zhou. Parameterized Complexity of the List Coloring Reconfiguration Problem with Graph Parameters. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 51:1-51:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{hatanaka_et_al:LIPIcs.MFCS.2017.51, author = {Hatanaka, Tatsuhiko and Ito, Takehiro and Zhou, Xiao}, title = {{Parameterized Complexity of the List Coloring Reconfiguration Problem with Graph Parameters}}, booktitle = {42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017)}, pages = {51:1--51:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-046-0}, ISSN = {1868-8969}, year = {2017}, volume = {83}, editor = {Larsen, Kim G. and Bodlaender, Hans L. and Raskin, Jean-Francois}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2017.51}, URN = {urn:nbn:de:0030-drops-81060}, doi = {10.4230/LIPIcs.MFCS.2017.51}, annote = {Keywords: combinatorial reconfiguration, fixed-parameter tractability, graph algorithm, list coloring, W\lbrack1\rbrack-hardness} }

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