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

Drawing a graph in the plane with as few crossings as possible is one of the central problems in graph drawing and computational geometry. Another option is to remove the smallest number of vertices or edges such that the remaining graph can be drawn without crossings. We study both problems in a book-embedding setting for ordered graphs, that is, graphs with a fixed vertex order. In this setting, the vertices lie on a straight line, called the spine, in the given order, and each edge must be drawn on one of several pages of a book such that every edge has at most a fixed number of crossings. In book embeddings, there is another way to reduce or avoid crossings; namely by using more pages. The minimum number of pages needed to draw an ordered graph without any crossings is its (fixed-vertex-order) page number.
We show that the page number of an ordered graph with n vertices and m edges can be computed in 2^m ⋅ n^𝒪(1) time. An 𝒪(log n)-approximation of this number can be computed efficiently. We can decide in 2^𝒪(d √k log (d+k)) ⋅ n^𝒪(1) time whether it suffices to delete k edges of an ordered graph to obtain a d-planar layout (where every edge crosses at most d other edges) on one page. As an additional parameter, we consider the size h of a hitting set, that is, a set of points on the spine such that every edge, seen as an open interval, contains at least one of the points. For h = 1, we can efficiently compute the minimum number of edges whose deletion yields fixed-vertex-order page number p. For h > 1, we give an XP algorithm with respect to h+p. Finally, we consider spine+t-track drawings, where some but not all vertices lie on the spine. The vertex order on the spine is given; we must map every vertex that does not lie on the spine to one of t tracks, each of which is a straight line on a separate page, parallel to the spine. In this setting, we can minimize in 2ⁿ ⋅ n^𝒪(1) time either the number of crossings or, if we disallow crossings, the number of tracks.

Akanksha Agrawal, Sergio Cabello, Michael Kaufmann, Saket Saurabh, Roohani Sharma, Yushi Uno, and Alexander Wolff. Eliminating Crossings in Ordered Graphs. In 19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 294, pp. 1:1-1:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{agrawal_et_al:LIPIcs.SWAT.2024.1, author = {Agrawal, Akanksha and Cabello, Sergio and Kaufmann, Michael and Saurabh, Saket and Sharma, Roohani and Uno, Yushi and Wolff, Alexander}, title = {{Eliminating Crossings in Ordered Graphs}}, booktitle = {19th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2024)}, pages = {1:1--1:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-318-8}, ISSN = {1868-8969}, year = {2024}, volume = {294}, editor = {Bodlaender, Hans L.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2024.1}, URN = {urn:nbn:de:0030-drops-200417}, doi = {10.4230/LIPIcs.SWAT.2024.1}, annote = {Keywords: Ordered graphs, book embedding, edge deletion, d-planar, hitting set} }

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Complete Volume

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

LIPIcs, Volume 226, FUN 2022, Complete Volume

11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 1-450, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@Proceedings{fraigniaud_et_al:LIPIcs.FUN.2022, title = {{LIPIcs, Volume 226, FUN 2022, Complete Volume}}, booktitle = {11th International Conference on Fun with Algorithms (FUN 2022)}, pages = {1--450}, 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}, URN = {urn:nbn:de:0030-drops-159693}, doi = {10.4230/LIPIcs.FUN.2022}, annote = {Keywords: LIPIcs, Volume 226, FUN 2022, Complete Volume} }

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Front Matter

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

Front Matter, Table of Contents, Preface, Conference Organization

11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 0:i-0:xii, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{fraigniaud_et_al:LIPIcs.FUN.2022.0, author = {Fraigniaud, Pierre and Uno, Yushi}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {11th International Conference on Fun with Algorithms (FUN 2022)}, pages = {0:i--0:xii}, 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.2022.0}, URN = {urn:nbn:de:0030-drops-159703}, doi = {10.4230/LIPIcs.FUN.2022.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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

We study the space reachable by rolling a 3D convex polyhedron on a 2D periodic tessellation in the xy-plane, where at every step a face of the polyhedron must coincide exactly with a tile of the tessellation it rests upon, and the polyhedron rotates around one of the incident edges of that face until the neighboring face hits the xy plane. If the whole plane can be reached by a sequence of such rolls, we call the polyhedron a plane roller for the given tessellation. We further classify polyhedra that reach a constant fraction of the plane, an infinite area but vanishing fraction of the plane, or a bounded area as hollow-plane rollers, band rollers, and bounded rollers respectively. We present a polynomial-time algorithm to determine the set of tiles in a given periodic tessellation reachable by a given polyhedron from a given starting position, which in particular determines the roller type of the polyhedron and tessellation. Using this algorithm, we compute the reachability for every regular-faced convex polyhedron on every regular-tiled (≤ 4)-uniform tessellation.

Akira Baes, Erik D. Demaine, Martin L. Demaine, Elizabeth Hartung, Stefan Langerman, Joseph O'Rourke, Ryuhei Uehara, Yushi Uno, and Aaron Williams. Rolling Polyhedra on Tessellations. In 11th International Conference on Fun with Algorithms (FUN 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 226, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{baes_et_al:LIPIcs.FUN.2022.6, author = {Baes, Akira and Demaine, Erik D. and Demaine, Martin L. and Hartung, Elizabeth and Langerman, Stefan and O'Rourke, Joseph and Uehara, Ryuhei and Uno, Yushi and Williams, Aaron}, title = {{Rolling Polyhedra on Tessellations}}, booktitle = {11th International Conference on Fun with Algorithms (FUN 2022)}, pages = {6:1--6:16}, 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.2022.6}, URN = {urn:nbn:de:0030-drops-159761}, doi = {10.4230/LIPIcs.FUN.2022.6}, annote = {Keywords: polyhedra, tilings} }

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

We propose a new kind of sliding-block puzzle, called Gourds, where the objective is to rearrange 1×2 pieces on a hexagonal grid board of 2n+1 cells with n pieces, using sliding, turning and pivoting moves. This puzzle has a single empty cell on a board and forms a natural extension of the 15-puzzle to include rotational moves. We analyze the puzzle and completely characterize the cases when the puzzle can always be solved. We also study the complexity of determining whether a given set of colored pieces can be placed on a colored hexagonal grid board with matching colors. We show this problem is NP-complete for arbitrarily many colors, but solvable in randomized polynomial time if the number of colors is a fixed constant.

Joep Hamersma, Marc van Kreveld, Yushi Uno, and Tom C. van der Zanden. Gourds: A Sliding-Block Puzzle with Turning. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 33:1-33:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{hamersma_et_al:LIPIcs.ISAAC.2020.33, author = {Hamersma, Joep and van Kreveld, Marc and Uno, Yushi and van der Zanden, Tom C.}, title = {{Gourds: A Sliding-Block Puzzle with Turning}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {33:1--33:16}, 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.2020.33}, URN = {urn:nbn:de:0030-drops-133773}, doi = {10.4230/LIPIcs.ISAAC.2020.33}, annote = {Keywords: computational complexity, divide-and-conquer, Hamiltonian cycle, puzzle game, (combinatorial) reconfiguration, sliding-block puzzle} }

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

In the economic activities,
the central bank has an important role to cover payments of banks,
when they are short of funds to clear their debts.
For this purpose, the central bank timely puts funds so that the economic activities go smooth.
Since payments in this mechanism are processed sequentially,
the total amount of funds put by the central bank critically
depends on the order of the payments.
Then an interest goes to the amount to prepare
if the order of the payments can be controlled by the central bank,
or if it is determined under the worst case scenario.
This motivates us to introduce a brand-new problem,
which we call the settlement fund circulation problem.
The problems are formulated as follows:
Let G=(V,A) be a directed multigraph with a vertex set V and an arc set A.
Each arc a\in A is endowed debt d(a)\ge 0,
and the debts are settled sequentially under a sequence \pi of arcs.
Each vertex v\in V is put fund in the amount of p_{\pi}(v)\ge 0
under the sequence.
The minimum/maximum settlement fund circulation problem (Min-SFC/Max-SFC)
in a given graph G with debts d: A\rightarrow \mathbb{R}_{+}\cup \{0\}
asks to find a bijection \pi:A\to \{1,2,\dots,|A|\}
that minimizes/maximizes the total funds \sum _{v\in V}p_{\pi }(v).
In this paper, we show that
both Min-SFC and Max-SFC are NP-hard;
in particular, Min-SFC is
(I) strongly NP-hard even if G is
(i) a multigraph with |V|=2 or (ii) a simple graph with treewidth at most two,and is (II) (not necessarily strongly) NP-hard for simple trees of diameter four,
while it is solvable in polynomial time for stars.
Also, we identify several polynomial time solvable cases for both problems.

Hitoshi Hayakawa, Toshimasa Ishii, Hirotaka Ono, and Yushi Uno. Settlement Fund Circulation Problem. In 28th International Symposium on Algorithms and Computation (ISAAC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 92, pp. 46:1-46:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{hayakawa_et_al:LIPIcs.ISAAC.2017.46, author = {Hayakawa, Hitoshi and Ishii, Toshimasa and Ono, Hirotaka and Uno, Yushi}, title = {{Settlement Fund Circulation Problem}}, booktitle = {28th International Symposium on Algorithms and Computation (ISAAC 2017)}, pages = {46:1--46:13}, 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.2017.46}, URN = {urn:nbn:de:0030-drops-82351}, doi = {10.4230/LIPIcs.ISAAC.2017.46}, annote = {Keywords: Fund settlement, Algorithm, Digraph, Scheduling} }

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

This paper studies a cooperative card game called Hanabi from an algorithmic combinatorial game theory viewpoint. The aim of the game is to play cards from 1 to n in increasing order (this has to be done independently in c different colors). Cards are drawn from a deck one by one. Drawn cards are either immediately played, discarded or stored for future use (overall each player can store up to h cards). The main feature of the game is that players know the cards their partners hold (but not theirs. This information must be shared through hints).
We introduce a simplified mathematical model of a single-player version of the game, and show several complexity results: the game is intractable in a general setting even if we forego with the hidden information aspect of the game. On the positive side, the game can be solved in linear time for some interesting restricted cases (i.e., for small values of h and c).

Jean-Francois Baffier, Man-Kwun Chiu, Yago Diez, Matias Korman, Valia Mitsou, André van Renssen, Marcel Roeloffzen, and Yushi Uno. Hanabi is NP-complete, Even for Cheaters who Look at Their Cards. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 4:1-4:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{baffier_et_al:LIPIcs.FUN.2016.4, author = {Baffier, Jean-Francois and Chiu, Man-Kwun and Diez, Yago and Korman, Matias and Mitsou, Valia and van Renssen, Andr\'{e} and Roeloffzen, Marcel and Uno, Yushi}, title = {{Hanabi is NP-complete, Even for Cheaters who Look at Their Cards}}, booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)}, pages = {4:1--4:17}, 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.2016.4}, URN = {urn:nbn:de:0030-drops-58644}, doi = {10.4230/LIPIcs.FUN.2016.4}, annote = {Keywords: algorithmic combinatorial game theory, sorting} }

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

We analyze the computational complexity of the popular computer games Threes!, 1024!, 2048 and many of their variants. For most known versions expanded to an m*n board, we show that it is NP-hard to decide whether a given starting position can be played to reach a specific (constant) tile value.

Stefan Langerman and Yushi Uno. Threes!, Fives, 1024!, and 2048 are Hard. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{langerman_et_al:LIPIcs.FUN.2016.22, author = {Langerman, Stefan and Uno, Yushi}, title = {{Threes!, Fives, 1024!, and 2048 are Hard}}, booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)}, pages = {22:1--22:14}, 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.2016.22}, URN = {urn:nbn:de:0030-drops-58661}, doi = {10.4230/LIPIcs.FUN.2016.22}, annote = {Keywords: algorithmic combinatorial game theory} }