Volume
LIPIcs, Volume 49
8th International Conference on Fun with Algorithms (FUN 2016)
-
Part of:
Series:
Leibniz International Proceedings in Informatics (LIPIcs)
Conference: International Conference on Fun with Algorithms (FUN)
Event
FUN 2016, June 8-10, 2016, La Maddalena, ItalyEditors
Publication Details
- published at: 2016-06-02
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-005-7
- DBLP: db/conf/fun/fun2016
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Document
Complete Volume
LIPIcs, Volume 49, FUN'16, Complete Volume
Abstract
LIPIcs, Volume 49, FUN'16, Complete Volume
Cite as
8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@Proceedings{demaine_et_al:LIPIcs.FUN.2016,
title = {{LIPIcs, Volume 49, FUN'16, Complete Volume}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
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},
URN = {urn:nbn:de:0030-drops-60579},
doi = {10.4230/LIPIcs.FUN.2016},
annote = {Keywords: Nonnumerical Algorithms and Problems, Discrete Mathematics, Complexity Measures and Classes}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{demaine_et_al:LIPIcs.FUN.2016.0,
author = {Demaine, Erik D. and Grandoni, Fabrizio},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {0:i--0:xvi},
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.2016.0},
URN = {urn:nbn:de:0030-drops-58624},
doi = {10.4230/LIPIcs.FUN.2016.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
2048 Without New Tiles Is Still Hard
Abstract
We study the computational complexity of a variant of the popular 2048 game in which no new tiles are generated after each move. As usual, instances are defined on rectangular boards of arbitrary size. We consider the natural decision problems of achieving a given constant tile value, score or number of moves. We also consider approximating the maximum achievable value for these three objectives. We prove all these problems are NP-hard by a reduction from 3SAT.
Furthermore, we consider potential extensions of these results to a similar variant of the Threes! game. To this end, we report on a peculiar motion pattern, that is not possible in 2048, which we found much harder to control by similar board designs.
Cite as
Ahmed Abdelkader, Aditya Acharya, and Philip Dasler. 2048 Without New Tiles Is Still Hard. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 1:1-1:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{abdelkader_et_al:LIPIcs.FUN.2016.1,
author = {Abdelkader, Ahmed and Acharya, Aditya and Dasler, Philip},
title = {{2048 Without New Tiles Is Still Hard}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {1:1--1:14},
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.2016.1},
URN = {urn:nbn:de:0030-drops-58858},
doi = {10.4230/LIPIcs.FUN.2016.1},
annote = {Keywords: Complexity of Games, 2048}
}
Document
Trainyard is NP-hard
Abstract
Recently, due to the widespread diffusion of smart-phones, mobile puzzle games have experienced a huge increase in their popularity. A successful puzzle has to be both captivating and challenging, and it has been suggested that this features are somehow related to their computational complexity. Indeed, many puzzle games - such as Mah-Jongg, Sokoban, Candy Crush, and 2048, to name a few - are known to be NP-hard.
In this paper we consider Trainyard: a popular mobile puzzle game whose goal is to get colored trains from their initial stations to suitable destination stations. We prove that the problem of determining whether there exists a solution to a given Trainyard level is NP. We also provide an implementation of our hardness reduction (see http://trainyard.isnphard.com).
Cite as
Matteo Almanza, Stefano Leucci, and Alessandro Panconesi. Trainyard is NP-hard. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 2:1-2:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{almanza_et_al:LIPIcs.FUN.2016.2,
author = {Almanza, Matteo and Leucci, Stefano and Panconesi, Alessandro},
title = {{Trainyard is NP-hard}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {2:1--2:14},
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.2016.2},
URN = {urn:nbn:de:0030-drops-58796},
doi = {10.4230/LIPIcs.FUN.2016.2},
annote = {Keywords: Complexity of Games, Trainyard}
}
Document
LOL: An Investigation into Cybernetic Humor, or: Can Machines Laugh?
Abstract
The mechanisms of humour have been the subject of much study and investigation, starting with and up to our days. Much of this work is based on literary theories, put forward by some of the most eminent philosophers and thinkers of all times, or medical theories, investigating the impact of humor on brain activity or behaviour. Recent functional neuroimaging studies, for instance, have investigated the process of comprehending and appreciating humor by examining functional activity in distinctive regions of brains stimulated by joke corpora. Yet, there is precious little work on the computational side, possibly due to the less hilarious nature of computer scientists as compared to men of letters and sawbones. In this paper, we set to investigate whether literary theories of humour can stand the test of algorithmic laughter. Or, in other words, we ask ourselves the vexed question: Can machines laugh?
We attempt to answer that question by testing whether an algorithm - namely, a neural network - can "understand" humour, and in particular whether it is possible to automatically identify abstractions that are predicted to be relevant by established literary theories about the mechanisms of humor. Notice that we do not focus here on distinguishing humorous from serious statements - a feat that is clearly way beyond the capabilities of the average human voter, not to mention the average machine - but rather on identifying the underlying mechanisms and triggers that are postulated to exist by literary theories, by verifying if similar mechanisms can be learned by machines.
Cite as
Davide Bacciu, Vincenzo Gervasi, and Giuseppe Prencipe. LOL: An Investigation into Cybernetic Humor, or: Can Machines Laugh?. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{bacciu_et_al:LIPIcs.FUN.2016.3,
author = {Bacciu, Davide and Gervasi, Vincenzo and Prencipe, Giuseppe},
title = {{LOL: An Investigation into Cybernetic Humor, or: Can Machines Laugh?}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {3:1--3:15},
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.2016.3},
URN = {urn:nbn:de:0030-drops-58823},
doi = {10.4230/LIPIcs.FUN.2016.3},
annote = {Keywords: deep learning, recurrent neural networks, dimensionality reduction algorithms}
}
Document
Hanabi is NP-complete, Even for Cheaters who Look at Their Cards
Abstract
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).
Cite as
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}
}
Document
Selenite Towers Move Faster Than Hanoï Towers, But Still Require Exponential Time
Abstract
The Hanoi Tower problem is a classic exercise in recursive programming: the solution has a simple recursive definition, and its complexity and the matching lower bound correspond to the solution of a simple recursive function (the solution is so simple that most students memorize it and regurgitate it at exams without truly understanding it). We describe how some minor change in the rules of the Hanoi Tower yields various increases of difficulty in the solution, so that to require a deeper mastery of recursion than the classical Hanoi Tower problem. In particular, we analyze the Selenite Tower problem, where just changing the insertion and extraction positions from the top to the middle of the tower results in a surprising increase in the intricacy of the solution: such a tower of n disks can be optimally moved in 3^(n/2) moves for n even (i.e. less than a Hanoi Tower of same height), via 5 recursive functions (or, equivalently, one recursion function with five states following three distinct patterns).
Cite as
Jérémy Barbay. Selenite Towers Move Faster Than Hanoï Towers, But Still Require Exponential Time. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{barbay:LIPIcs.FUN.2016.5,
author = {Barbay, J\'{e}r\'{e}my},
title = {{Selenite Towers Move Faster Than Hano\"{i} Towers, But Still Require Exponential Time}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {5:1--5:20},
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.2016.5},
URN = {urn:nbn:de:0030-drops-58729},
doi = {10.4230/LIPIcs.FUN.2016.5},
annote = {Keywords: Br\"{a}hma tower, Disk Pile, Hanoi Tower, Levitating Tower, Recursivity}
}
Document
Algorithms and Insights for RaceTrack
Abstract
We discuss algorithmic issues on the well-known paper-and-pencil game RaceTrack. On a very simple track called Indianapolis, we introduce the problem and simple approaches, that will be gradually refined. We present and experimentally evaluate efficient algorithms for single player scenarios. We also consider a variant where the parts of the track are known as soon as they become visible during the race.
Cite as
Michael A. Bekos, Till Bruckdorfer, Henry Förster, Michael Kaufmann, Simon Poschenrieder, and Thomas Stüber. Algorithms and Insights for RaceTrack. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 6:1-6:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{bekos_et_al:LIPIcs.FUN.2016.6,
author = {Bekos, Michael A. and Bruckdorfer, Till and F\"{o}rster, Henry and Kaufmann, Michael and Poschenrieder, Simon and St\"{u}ber, Thomas},
title = {{Algorithms and Insights for RaceTrack}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {6:1--6:14},
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.2016.6},
URN = {urn:nbn:de:0030-drops-58818},
doi = {10.4230/LIPIcs.FUN.2016.6},
annote = {Keywords: Racetrack, State-graph, complexity}
}
Document
Resource Optimization for Program Committee Members: A Subreview Article
Abstract
This paper formalizes a resource-allocation problem that is all too familiar to the seasoned program-committee member. For each submission j that the PC member has the honor of reviewing, there is a choice. The PC member can spend the time to review submission j in detail on his/her own at a cost of C_i. Alternatively, the PC member can spend the time to identify and contact peers, hoping to recruit them as subreviewers, at a cost of 1 per subreviewer. These potential subreviewers have a certain probability of rejecting each review request, and this probability increases as time goes on. Once the PC member runs out of time or unasked experts, he/she is forced to review the paper without outside assistance.
This paper gives optimal solutions to several variations of the scheduling-reviewers problem. Most of the solutions from this paper are based on an iterated log function of C_i. In particular, with k rounds, the optimal solution sends the k-iterated log of C_i requests in the first round, the (k-1)-iterated log in the second round, and so forth. One of the contributions of this paper is solving this problem exactly, even when rejection probabilities may increase.
Naturally, PC members must make an integral number of subreview requests. This paper gives, as an intermediate result, a linear-time algorithm to transform the artificial problem in which one can send fractional requests into the less-artificial problem in which one sends an integral number of requests. Finally, this paper considers the case where the PC member knows nothing about the probability that a potential subreviewer agrees to review the paper. This paper gives an approximation algorithm for this case, whose bounds improve as the number of rounds increases.
Cite as
Michael A. Bender, Samuel McCauley, Bertrand Simon, Shikha Singh, and Frédéric Vivien. Resource Optimization for Program Committee Members: A Subreview Article. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 7:1-7:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{bender_et_al:LIPIcs.FUN.2016.7,
author = {Bender, Michael A. and McCauley, Samuel and Simon, Bertrand and Singh, Shikha and Vivien, Fr\'{e}d\'{e}ric},
title = {{Resource Optimization for Program Committee Members: A Subreview Article}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {7:1--7:20},
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.2016.7},
URN = {urn:nbn:de:0030-drops-58872},
doi = {10.4230/LIPIcs.FUN.2016.7},
annote = {Keywords: Scheduling, Delegation, Subreviews}
}
Document
Physical Zero-Knowledge Proofs for Akari, Takuzu, Kakuro and KenKen
Abstract
Akari, Takuzu, Kakuro and KenKen are logic games similar to Sudoku. In Akari, a labyrinth on a grid has to be lit by placing lanterns, respecting various constraints. In Takuzu a grid has to be filled with 0's and 1's, while respecting certain constraints. In Kakuro a grid has to be filled with numbers such that the sums per row and column match given values; similarly in KenKen a grid has to be filled with numbers such that in given areas the product, sum, difference or quotient equals a given value. We give physical algorithms to realize zero-knowledge proofs for these games which allow a player to show that he knows a solution without revealing it. These interactive proofs can be realized with simple office material as they only rely on cards and envelopes. Moreover, we formalize our algorithms and prove their security.
Cite as
Xavier Bultel, Jannik Dreier, Jean-Guillaume Dumas, and Pascal Lafourcade. Physical Zero-Knowledge Proofs for Akari, Takuzu, Kakuro and KenKen. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{bultel_et_al:LIPIcs.FUN.2016.8,
author = {Bultel, Xavier and Dreier, Jannik and Dumas, Jean-Guillaume and Lafourcade, Pascal},
title = {{Physical Zero-Knowledge Proofs for Akari, Takuzu, Kakuro and KenKen}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {8:1--8:20},
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.2016.8},
URN = {urn:nbn:de:0030-drops-58693},
doi = {10.4230/LIPIcs.FUN.2016.8},
annote = {Keywords: Physical Cryptography, ZKP, Games, Akari, Kakuro, KenKen, Takuzu}
}
Document
Analyzing and Comparing On-Line News Sources via (Two-Layer) Incremental Clustering
Abstract
In this paper, we analyse the contents of the web site of two Italian press agencies and of four of the most popular Italian newspapers, in order to answer questions such as what are the most relevant news, what is the average life of news, and how much different are different sites. To this aim, we have developed a web-based application which hourly collects the articles in the main column of the six web sites, implements an incremental clustering algorithm for grouping the articles into news, and finally allows the user to see the answer to the above questions. We have also designed and implemented a two-layer modification of the incremental clustering algorithm and executed some preliminary experimental evaluation of this modification: it turns out that the two-layer clustering is extremely efficient in terms of time performances, and it has quite good performances in terms of precision and recall.
Cite as
Francesco Cambi, Pierluigi Crescenzi, and Linda Pagli. Analyzing and Comparing On-Line News Sources via (Two-Layer) Incremental Clustering. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{cambi_et_al:LIPIcs.FUN.2016.9,
author = {Cambi, Francesco and Crescenzi, Pierluigi and Pagli, Linda},
title = {{Analyzing and Comparing On-Line News Sources via (Two-Layer) Incremental Clustering}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {9:1--9:14},
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.2016.9},
URN = {urn:nbn:de:0030-drops-58777},
doi = {10.4230/LIPIcs.FUN.2016.9},
annote = {Keywords: text mining, incremental clustering, on-line news}
}
Document
Spy-Game on Graphs
Abstract
We define and study the following two-player game on a graph G. Let k in N^*. A set of k guards is occupying some vertices of G while one spy is standing at some node. At each turn, first the spy may move along at most s edges, where s in N^* is his speed. Then, each guard may move along one edge. The spy and the guards may occupy same vertices. The spy has to escape the surveillance of the guards, i.e., must reach a vertex at distance more than d in N (a predefined distance) from every guard. Can the spy win against k guards? Similarly, what is the minimum distance d such that k guards may ensure that at least one of them remains at distance at most d from the spy?
This game generalizes two well-studied games: Cops and robber games (when s=1) and Eternal Dominating Set (when s is unbounded).
We consider the computational complexity of the problem, showing that it is NP-hard and that it is PSPACE-hard in DAGs. Then, we establish tight tradeoffs between the number of guards and the required distance d when G is a path or a cycle. Our main result is that there exists beta>0 such that Omega(n^{1+beta}) guards are required to win in any n*n grid.
Cite as
Nathann Cohen, Mathieu Hilaire, Nícolas A. Martins, Nicolas Nisse, and Stéphane Pérennes. Spy-Game on Graphs. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 10:1-10:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{cohen_et_al:LIPIcs.FUN.2016.10,
author = {Cohen, Nathann and Hilaire, Mathieu and Martins, N{\'\i}colas A. and Nisse, Nicolas and P\'{e}rennes, St\'{e}phane},
title = {{Spy-Game on Graphs}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {10:1--10:16},
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.2016.10},
URN = {urn:nbn:de:0030-drops-58782},
doi = {10.4230/LIPIcs.FUN.2016.10},
annote = {Keywords: graph, two-player games, cops and robber games, complexity}
}
Document
The Complexity of Snake
Abstract
Snake and Nibbler are two well-known video games in which a snake slithers through a maze and grows as it collects food. During this process, the snake must avoid any collision with its tail. Various goals can be associated with these video games, such as avoiding the tail as long as possible, or collecting a certain amount of food, or reaching some target location. Unfortunately, like many other motion-planning problems, even very restricted variants are computationally intractable. In particular, we prove the NP--hardness of collecting all food on solid grid graphs; as well as its PSPACE-completeness on general grid graphs. Moreover, given an initial and a target configuration of the snake, moving from one configuration to the other is PSPACE-complete, even on grid graphs without food, or with an initially short snake.
Our results make use of the nondeterministic constraint logic framework by Hearn and Demaine, which has been used to analyze the computational complexity of many games and puzzles. We extend this framework for the analysis of puzzles whose initial state is chosen by the player.
Cite as
Marzio De Biasi and Tim Ophelders. The Complexity of Snake. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 11:1-11:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{debiasi_et_al:LIPIcs.FUN.2016.11,
author = {De Biasi, Marzio and Ophelders, Tim},
title = {{The Complexity of Snake}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {11:1--11:13},
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.2016.11},
URN = {urn:nbn:de:0030-drops-58841},
doi = {10.4230/LIPIcs.FUN.2016.11},
annote = {Keywords: Games, Puzzles, Motion Planning, Nondeterministic Constraint Logic, PSPACE}
}
Document
The Fewest Clues Problem
Abstract
When analyzing the computational complexity of well-known puzzles, most papers consider the algorithmic challenge of solving a given instance of (a generalized form of) the puzzle. We take a different approach by analyzing the computational complexity of designing a "good" puzzle. We assume a puzzle maker designs part of an instance, but before publishing it, wants to ensure that the puzzle has a unique solution. Given a puzzle, we introduce the FCP (fewest clues problem) version of the problem:
Given an instance to a puzzle, what is the minimum number of clues we must add in order to make the instance uniquely solvable?
We analyze this question for the Nikoli puzzles Sudoku, Shakashaka, and Akari. Solving these puzzles is NP-complete, and we show their FCP versions are Sigma_2^P-complete. Along the way, we show that the FCP versions of 3SAT, 1-in-3SAT, Triangle Partition, Planar 3SAT, and Latin Square are all Sigma_2^P-complete. We show that even problems in P have difficult FCP versions, sometimes even Sigma_2^P-complete, though "closed under cluing" problems are in the (presumably) smaller class NP; for example, FCP 2SAT is NP-complete.
Cite as
Erik D. Demaine, Fermi Ma, Ariel Schvartzman, Erik Waingarten, and Scott Aaronson. The Fewest Clues Problem. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 12:1-12:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{demaine_et_al:LIPIcs.FUN.2016.12,
author = {Demaine, Erik D. and Ma, Fermi and Schvartzman, Ariel and Waingarten, Erik and Aaronson, Scott},
title = {{The Fewest Clues Problem}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {12:1--12:12},
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.2016.12},
URN = {urn:nbn:de:0030-drops-58654},
doi = {10.4230/LIPIcs.FUN.2016.12},
annote = {Keywords: computational complexity, pencil-and-paper puzzles, hardness reductions}
}
Document
Super Mario Bros. is Harder/Easier Than We Thought
Abstract
Mario is back! In this sequel, we prove that solving a generalized level of Super Mario Bros. is PSPACE-complete, strengthening the previous NP-hardness result (FUN 2014). Both our PSPACE-hardness and the previous NP-hardness use levels of arbitrary dimensions and require either arbitrarily large screens or a game engine that remembers the state of off-screen sprites. We also analyze the complexity of the less general case where the screen size is constant, the number of on-screen sprites is constant, and the game engine forgets the state of everything substantially off-screen, as in most, if not all, Super Mario Bros. video games. In this case we prove that the game is solvable in polynomial time, assuming levels are explicitly encoded; on the other hand, if levels can be represented using run-length encoding, then the problem is weakly NP-hard (even if levels have only constant height, as in the video games). All of our hardness proofs are also resilient to known glitches in Super Mario Bros., unlike the previous NP-hardness proof.
Cite as
Erik D. Demaine, Giovanni Viglietta, and Aaron Williams. Super Mario Bros. is Harder/Easier Than We Thought. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{demaine_et_al:LIPIcs.FUN.2016.13,
author = {Demaine, Erik D. and Viglietta, Giovanni and Williams, Aaron},
title = {{Super Mario Bros. is Harder/Easier Than We Thought}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {13:1--13:14},
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.2016.13},
URN = {urn:nbn:de:0030-drops-58802},
doi = {10.4230/LIPIcs.FUN.2016.13},
annote = {Keywords: video games, computational complexity, PSPACE}
}
Document
A Rupestrian Algorithm
Abstract
Deciphering recently discovered cave paintings by the Astracinca, an egalitarian leaderless society flourishing in the 3rd millennium BCE, we present and analyze their shamanic ritual for forming new colonies. This ritual can actually be used by systems of anonymous mobile finite-state computational entities located and operating in a grid to solve the line recovery problem, a task that has both self-assembly and flocking requirements. The protocol is totally decentralized, fully concurrent, provably correct, and time optimal.
Cite as
Giuseppe A. Di Luna, Paola Flocchini, Giuseppe Prencipe, Nicola Santoro, and Giovanni Viglietta. A Rupestrian Algorithm. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 14:1-14:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{diluna_et_al:LIPIcs.FUN.2016.14,
author = {Di Luna, Giuseppe A. and Flocchini, Paola and Prencipe, Giuseppe and Santoro, Nicola and Viglietta, Giovanni},
title = {{A Rupestrian Algorithm}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {14:1--14:20},
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.2016.14},
URN = {urn:nbn:de:0030-drops-58751},
doi = {10.4230/LIPIcs.FUN.2016.14},
annote = {Keywords: mobile finite-state machines, self-healing distributed algorithms}
}
Document
Building a Better Mouse Maze
Abstract
Mouse Maze is a Flash game about Squeaky, a mouse who has to navigate a subset of the grid using a simple deterministic rule, which naturally generalises to a game on arbitrary graphs with some interesting chaotic dynamics. We present the results of some evolutionary algorithms which generate graphs which effectively trap Squeaky in the maze for long periods of time, and some theoretical results on how long he can be trapped. We then discuss what would happen to Squeaky if he couldn't count, and present some open problems in the area.
Cite as
Jessica Enright and John D. Faben. Building a Better Mouse Maze. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 15:1-15:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{enright_et_al:LIPIcs.FUN.2016.15,
author = {Enright, Jessica and Faben, John D.},
title = {{Building a Better Mouse Maze}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {15:1--15:12},
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.2016.15},
URN = {urn:nbn:de:0030-drops-58743},
doi = {10.4230/LIPIcs.FUN.2016.15},
annote = {Keywords: graph, evolutionary, genetic algorithm, traversal}
}
Document
Recognizing a DOG is Hard, But Not When It is Thin and Unit
Abstract
We define the notion of disk-obedience for a set of disks in the plane and give results for diskobedient graphs (DOGs), which are disk intersection graphs (DIGs) that admit a planar embedding with vertices inside the corresponding disks. We show that in general it is hard to recognize a DOG, but when the DIG is thin and unit (i.e., when the disks are unit disks), it can be done in linear time.
Cite as
William Evans, Mereke van Garderen, Maarten Löffler, and Valentin Polishchuk. Recognizing a DOG is Hard, But Not When It is Thin and Unit. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 16:1-16:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{evans_et_al:LIPIcs.FUN.2016.16,
author = {Evans, William and van Garderen, Mereke and L\"{o}ffler, Maarten and Polishchuk, Valentin},
title = {{Recognizing a DOG is Hard, But Not When It is Thin and Unit}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {16:1--16:12},
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.2016.16},
URN = {urn:nbn:de:0030-drops-58671},
doi = {10.4230/LIPIcs.FUN.2016.16},
annote = {Keywords: graph drawing, planar graphs, disk intersection graphs}
}
Document
Counting Circles Without Computing Them
Abstract
In this paper we engineer a fast algorithm to count the number of triangles defined by three lines out of a set of n lines whose circumcircle contains the origin. The trick is not to compute any triangles or circles.
Cite as
Rudolf Fleischer. Counting Circles Without Computing Them. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 17:1-17:7, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{fleischer:LIPIcs.FUN.2016.17,
author = {Fleischer, Rudolf},
title = {{Counting Circles Without Computing Them}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {17:1--17:7},
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.2016.17},
URN = {urn:nbn:de:0030-drops-58767},
doi = {10.4230/LIPIcs.FUN.2016.17},
annote = {Keywords: lines arrangement, triangle, circumcircle, inscribed angle theorem}
}
Document
Large Peg-Army Maneuvers
Abstract
Despite its long history, the classical game of peg solitaire continues to attract the attention of the scientific community. In this paper, we consider two problems with an algorithmic flavour which are related with this game, namely Solitaire-Reachability and Solitaire-Army. In the first one, we show that deciding whether there is a sequence of jumps which allows a given initial configuration of pegs to reach a target position is NP-complete. Regarding Solitaire-Army, the aim is to successfully deploy an army of pegs in a given region of the board in order to reach a target position. By solving an auxiliary problem with relaxed constraints, we are able to answer some open questions raised by Csakany and Juhasz (Mathematics Magazine, 2000).
Cite as
Luciano Gualà, Stefano Leucci, Emanuele Natale, and Roberto Tauraso. Large Peg-Army Maneuvers. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{guala_et_al:LIPIcs.FUN.2016.18,
author = {Gual\`{a}, Luciano and Leucci, Stefano and Natale, Emanuele and Tauraso, Roberto},
title = {{Large Peg-Army Maneuvers}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {18:1--18:15},
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.2016.18},
URN = {urn:nbn:de:0030-drops-58709},
doi = {10.4230/LIPIcs.FUN.2016.18},
annote = {Keywords: Complexity of Games, Solitaire Army}
}
Document
Loopless Gray Code Enumeration and the Tower of Bucharest
Abstract
We give new algorithms for generating all n-tuples over an alphabet of m letters, changing only one letter at a time (Gray codes). These algorithms are based on the connection with variations of the Towers of Hanoi game. Our algorithms are loopless, in the sense that the next change can be determined in a constant number of steps, and they can be implemented in hardware. We also give another family of loopless algorithms that is based on the idea of working ahead and saving the work in a buffer.
Cite as
Felix Herter and Günter Rote. Loopless Gray Code Enumeration and the Tower of Bucharest. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{herter_et_al:LIPIcs.FUN.2016.19,
author = {Herter, Felix and Rote, G\"{u}nter},
title = {{Loopless Gray Code Enumeration and the Tower of Bucharest}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {19:1--19:19},
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.2016.19},
URN = {urn:nbn:de:0030-drops-58863},
doi = {10.4230/LIPIcs.FUN.2016.19},
annote = {Keywords: Tower of Hanoi, Gray code, enumeration, loopless generation}
}
Document
Convex Configurations on Nana-kin-san Puzzle
Abstract
We investigate a silhouette puzzle that is recently developed based on the golden ratio. Traditional silhouette puzzles are based on a simple tile. For example, the tangram is based on isosceles right triangles; that is, each of seven pieces is formed by gluing some identical isosceles right triangles. Using the property, we can analyze it by hand, that is, without computer. On the other hand, if each piece has no special property, it is quite hard even using computer since we have to handle real numbers without numerical errors during computation. The new silhouette puzzle is between them; each of seven pieces is not based on integer length and right angles, but based on golden ratio, which admits us to represent these seven pieces in some nontrivial way. Based on the property, we develop an algorithm to handle the puzzle, and our algorithm succeeded to enumerate all convex shapes that can be made by the puzzle pieces.
It is known that the tangram and another classic silhouette puzzle known as Sei-shonagon chie no ita can form 13 and 16 convex shapes, respectively. The new puzzle, Nana-kin-san puzzle, admits to form 62 different convex shapes.
Cite as
Takashi Horiyama, Ryuhei Uehara, and Haruo Hosoya. Convex Configurations on Nana-kin-san Puzzle. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 20:1-20:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{horiyama_et_al:LIPIcs.FUN.2016.20,
author = {Horiyama, Takashi and Uehara, Ryuhei and Hosoya, Haruo},
title = {{Convex Configurations on Nana-kin-san Puzzle}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {20:1--20:14},
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.2016.20},
URN = {urn:nbn:de:0030-drops-58730},
doi = {10.4230/LIPIcs.FUN.2016.20},
annote = {Keywords: silhouette puzzles, nana-kin-san puzzle, enumeration algorithm, convex polygon}
}
Document
How to Solve the Cake-Cutting Problem in Sublinear Time
Abstract
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.
Cite as
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}
}
Document
Threes!, Fives, 1024!, and 2048 are Hard
Abstract
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.
Cite as
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}
}
Document
An Arithmetic for Rooted Trees
Abstract
We propose a new arithmetic for non-empty rooted unordered trees simply called trees. After discussing tree representation and enumeration, we define the operations of tree addition, multiplication, and stretch, prove their properties, and show that all trees can be generated from a starting tree of one vertex. We then show how a given tree can be obtained as the sum or product of two trees, thus defining prime trees with respect to addition and multiplication. In both cases we show how primality can be decided in time polynomial in the number of vertices and prove that factorization is unique.
We then define negative trees and suggest dealing with tree equations, giving some preliminary examples. Finally we comment on how our arithmetic might be useful, and discuss preceding studies that have some relations with ours. The parts of this work that do not concur to an immediate illustration of our proposal, including formal proofs, are reported in the Appendix.
To the best of our knowledge our proposal is completely new and can be largely modified in cooperation with the readers. To the ones of his age the author suggests that "many roads must be walked down before we call it a theory".
Cite as
Fabrizio Luccio. An Arithmetic for Rooted Trees. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 23:1-23:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{luccio:LIPIcs.FUN.2016.23,
author = {Luccio, Fabrizio},
title = {{An Arithmetic for Rooted Trees}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {23:1--23:14},
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.2016.23},
URN = {urn:nbn:de:0030-drops-58687},
doi = {10.4230/LIPIcs.FUN.2016.23},
annote = {Keywords: Arithmetic, Rooted tree, Prime tree, Tree equation}
}
Document
Two Dots is NP-complete
Abstract
Two Dots is a popular single-player puzzle video game for iOS and Android. In its simplest form, the game consists of a board with dots of different colors, and a valid move consists of connecting a sequence of adjacent dots of the same color. We say that dots engaged in a move are "hit" by the player. After every move, the connected dots disappear, and the void is filled by new dots (the entire board shifts downwards and new dots appear on top). Typically the game provides a limited number of moves and varying goals (such as hitting a required number of dots of a particular color). We show that the perfect information version of the game (where the sequence of incoming dots is known) is NP-complete, even for fairly restricted goal types.
Cite as
Neeldhara Misra. Two Dots is NP-complete. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 24:1-24:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{misra:LIPIcs.FUN.2016.24,
author = {Misra, Neeldhara},
title = {{Two Dots is NP-complete}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {24:1--24:12},
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.2016.24},
URN = {urn:nbn:de:0030-drops-58831},
doi = {10.4230/LIPIcs.FUN.2016.24},
annote = {Keywords: combinatorial game theory, NP-complete, perfect information, puzzle}
}
Document
This House Proves That Debating is Harder Than Soccer
Abstract
During the last twenty years, a lot of research was conducted on the sport elimination problem: Given a sports league and its remaining matches, we have to decide whether a given team can still possibly win the competition, i.e., place first in the league at the end. Previously, the computational complexity of this problem was investigated only for games with two participating teams per game. In this paper we consider Debating Tournaments and Debating Leagues in the British Parliamentary format, where four teams are participating in each game. We prove that it is NP-hard to decide whether a given team can win a Debating League, even if at most two matches are remaining for each team. This contrasts settings like football where two teams play in each game since there this case is still polynomial time solvable. We prove our result even for a fictitious restricted setting with only three teams per game. On the other hand, for the common setting of Debating Tournaments we show that this problem is fixed parameter tractable if the parameter is the number of remaining rounds k. This also holds for the practically very important question of whether a team can still qualify for the knock-out phase of the tournament and the combined parameter k+b where b denotes the threshold rank for qualifying. Finally, we show that the latter problem is polynomial time solvable for any constant k and arbitrary values b that are part of the input.
Cite as
Stefan Neumann and Andreas Wiese. This House Proves That Debating is Harder Than Soccer. In 8th International Conference on Fun with Algorithms (FUN 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 49, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
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@InProceedings{neumann_et_al:LIPIcs.FUN.2016.25,
author = {Neumann, Stefan and Wiese, Andreas},
title = {{This House Proves That Debating is Harder Than Soccer}},
booktitle = {8th International Conference on Fun with Algorithms (FUN 2016)},
pages = {25:1--25:14},
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.2016.25},
URN = {urn:nbn:de:0030-drops-58716},
doi = {10.4230/LIPIcs.FUN.2016.25},
annote = {Keywords: complexity, elimination games, soccer, debating}
}