8th International Conference on Fun with Algorithms (FUN 2016), FUN 2016, June 8-10, 2016, La Maddalena, Italy
FUN 2016
June 8-10, 2016
La Maddalena, Italy
International Conference on Fun with Algorithms
FUN
http://erikdemaine.org/events/?prefix=FUN
https://dblp.org/db/conf/fun
Leibniz International Proceedings in Informatics
LIPIcs
https://www.dagstuhl.de/dagpub/1868-8969
https://dblp.org/db/series/lipics
1868-8969
Erik D.
Demaine
Erik D. Demaine
Fabrizio
Grandoni
Fabrizio Grandoni
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
49
2016
978-3-95977-005-7
https://www.dagstuhl.de/dagpub/978-3-95977-005-7
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter, Table of Contents, Preface, Conference Organization
Front Matter
Table of Contents
Preface
Conference Organization
0:i-0:xvi
Front Matter
Erik D.
Demaine
Erik D. Demaine
Fabrizio
Grandoni
Fabrizio Grandoni
10.4230/LIPIcs.FUN.2016.0
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
2048 Without New Tiles Is Still Hard
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.
Complexity of Games
2048
1:1-1:14
Regular Paper
Ahmed
Abdelkader
Ahmed Abdelkader
Aditya
Acharya
Aditya Acharya
Philip
Dasler
Philip Dasler
10.4230/LIPIcs.FUN.2016.1
Ahmed Abdelkader. 2048 gadgets. URL: http://cs.umd.edu/~akader/projects/2048/index.html.
http://cs.umd.edu/~akader/projects/2048/index.html
Ahmed Abdelkader, Aditya Acharya, and Philip Dasler. 2048 is NP-Complete. CGYRF, 2015.
Ahmed Abdelkader, Aditya Acharya, and Philip Dasler. On the Complexity of Slide-and-Merge Games. CoRR, abs/1501.03837, 2015. URL: http://arxiv.org/abs/1501.03837.
http://arxiv.org/abs/1501.03837
Christopher Chen. 2048 is in NP. URL: http://blog.openendings.net/2014/03/2048-is-in-np.html.
http://blog.openendings.net/2014/03/2048-is-in-np.html
Erik D. Demaine, Martin L. Demaine, and Joseph O'Rourke. PushPush is NP-hard in 2D. CoRR, cs.CG/0001019, 2000. URL: http://arxiv.org/abs/cs.CG/0001019.
http://arxiv.org/abs/cs.CG/0001019
Luciano Guala, Stefano Leucci, and Emanuele Natale. Bejeweled, Candy Crush and other Match-Three Games are (NP-)Hard. In 2014 IEEE Conference on Computational Intelligence and Games, pages 1-8. IEEE, 2014.
Stefan Langerman and Yushi Uno. Threes!, Fives, 1024!, and 2048 are Hard. CoRR, abs/1505.04274, 2015. URL: http://arxiv.org/abs/1505.04274.
http://arxiv.org/abs/1505.04274
Stefan Langerman and Yushi Uno. Threes!, Fives, 1024!, and 2048 are Hard. In 8th International Conference on Fun with Algorithms (FUN 2016), volume 49 of Leibniz International Proceedings in Informatics (LIPIcs), pages 22:1-22:14, 2016.
Rahul Mehta. 2048 is (PSPACE) Hard, but Sometimes Easy. CoRR, abs/1408.6315, 2014. URL: http://arxiv.org/abs/1408.6315.
http://arxiv.org/abs/1408.6315
Creative Commons Attribution 3.0 Unported license
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Trainyard is NP-hard
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).
Complexity of Games
Trainyard
2:1-2:14
Regular Paper
Matteo
Almanza
Matteo Almanza
Stefano
Leucci
Stefano Leucci
Alessandro
Panconesi
Alessandro Panconesi
10.4230/LIPIcs.FUN.2016.2
Greg Aloupis, Erik D. Demaine, Alan Guo, and Giovanni Viglietta. Classic nintendo games are (computationally) hard. Theoretical Computer Science, 586:135-160, 2015. URL: http://dx.doi.org/10.1016/j.tcs.2015.02.037.
http://dx.doi.org/10.1016/j.tcs.2015.02.037
Casual Games Association. Casual games sector report: Towards the global games market in 2017. http://www.casualconnect.org/education.html. Accessed: 2016-02-22.
http://www.casualconnect.org/education.html
Anne Condon, Joan Feigenbaum, Carsten Lund, and Peter W. Shor. Random debaters and the hardness of approximating stochastic functions. SIAM Journal on Computing, 26(2):369-400, 1997. URL: http://dx.doi.org/10.1137/S0097539793260738.
http://dx.doi.org/10.1137/S0097539793260738
Joseph Culberson. Sokoban is pspace-complete. In Proceedings of the 1st International Conference on Fun with Algorithms (FUN'98), 1998, volume 4, pages 65-76, 1998.
David Eppstein. Computational complexity of games and puzzles. https://www.ics.uci.edu/~eppstein/cgt/hard.html. Accessed: 2016-02-22.
https://www.ics.uci.edu/~eppstein/cgt/hard.html
Gary William Flake and Eric B. Baum. Rush hour is PSPACE-complete, or "Why you should generously tip parking lot attendants". Theoretical Computer Science, 270(1-2):895-911, 2002. URL: http://dx.doi.org/10.1016/S0304-3975(01)00173-6.
http://dx.doi.org/10.1016/S0304-3975(01)00173-6
Luciano Gualà, Stefano Leucci, and Emanuele Natale. Bejeweled, candy crush and other match-three games are (NP-)hard. In Proceedings of the 2014 IEEE Conference on Computational Intelligence and Games (CIG'14), 2014, pages 1-8, 2014. URL: http://dx.doi.org/10.1109/CIG.2014.6932866.
http://dx.doi.org/10.1109/CIG.2014.6932866
Robert A. Hearn and Erik D. Demaine. Games, puzzles, and computation. CRC Press, 2009.
Graham Kendall, Andrew J. Parkes, and Kristian Spoerer. A survey of NP-complete puzzles. ICGA Journal, 31(1):13-34, 2008.
Rahul Mehta. 2048 is (PSPACE) hard, but sometimes easy. CoRR, abs/1408.6315, 2014.
Daniel Ratner and Manfred K. Warmuth. Nxn puzzle and related relocation problem. Journal of Symbolic Computation, 10(2):111-138, 1990. URL: http://dx.doi.org/10.1016/S0747-7171(08)80001-6.
http://dx.doi.org/10.1016/S0747-7171(08)80001-6
Matt Rix. The story so far. http://struct.ca/2010/the-story-so-far/. Accessed: 2016-02-22.
http://struct.ca/2010/the-story-so-far/
Matt Rix. Trains on paper. http://struct.ca/2010/trains-on-paper/. Accessed: 2016-02-22.
http://struct.ca/2010/trains-on-paper/
Matt Rix. The week that was. http://struct.ca/2010/the-week-that-was/. Accessed: 2016-02-22.
http://struct.ca/2010/the-week-that-was/
Giovanni Viglietta. Gaming is a hard job, but someone has to do it! Theory Computing Systems, 54(4):595-621, 2014. URL: http://dx.doi.org/10.1007/s00224-013-9497-5.
http://dx.doi.org/10.1007/s00224-013-9497-5
Creative Commons Attribution 3.0 Unported license
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LOL: An Investigation into Cybernetic Humor, or: Can Machines Laugh?
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.
deep learning
recurrent neural networks
dimensionality reduction algorithms
3:1-3:15
Regular Paper
Davide
Bacciu
Davide Bacciu
Vincenzo
Gervasi
Vincenzo Gervasi
Giuseppe
Prencipe
Giuseppe Prencipe
10.4230/LIPIcs.FUN.2016.3
Aristotle. Poetics, volume two: On Comedy. IV century BCE. Last surviving copy reportedly lost in the fire of the Abbey’s Library in 1327, according to [6].
Salvatore Attardo. Linguistic Theories of Humor. De Gruyter Mouton, Berlin, 1994.
Yoshua Bengio, Patrice Simard, and Paolo Frasconi. Learning long-term dependencies with gradient descent is difficult. Neural Networks, IEEE Transactions on, 5(2):157-166, 1994.
Yu-Chen Chan. Emotional structure of jokes: A corpus-based investigation. Bio-medical materials and engineering, 24(6):3083-3090, 2014.
Yu-Chen Chan and Joseph P Lavallee. Temporo-parietal and fronto-parietal lobe contributions to theory of mind and executive control: an fMRI study of verbal jokes. Frontiers in psychology, 6, 2015.
Umberto Eco. Il nome della rosa. Bompiani, 1980.
Paolo Frasconi, Marco Gori, and Alessandro Sperduti. A general framework for adaptive processing of data structures. Neural Networks, IEEE Transactions on, 9(5):768-786, 1998.
Lisa Friedland and James Allan. Joke retrieval: Recognizing the same joke told differently. In Proceedings of the 17th ACM Conference on Information and Knowledge Management, CIKM'08, pages 883-892, New York, NY, USA, 2008. ACM.
Alex Graves. Generating sequences with recurrent neural networks. CoRR, abs/1308.0850, 2013. URL: http://arxiv.org/abs/1308.0850.
http://arxiv.org/abs/1308.0850
Alex Graves and Jürgen Schmidhuber. Framewise phoneme classification with bidirectional lstm and other neural network architectures. Neural Networks, 18(5):602-610, 2005.
Ulrich Günther. What is in a laugh? Humour, jokes and laughter in the conversational corpus of the BNC. PhD thesis, Albert-Ludwigs-Universität, Freiburg, 2003.
Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation, 9(8):1735-1780, 1997.
Laurens Van Der Maaten. Accelerating t-sne using tree-based algorithms. The Journal of Machine Learning Research, 15(1):3221-3245, 2014.
Laurens Van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of Machine Learning Research, 9(2579-2605):85, 2008.
Wendel Wallach and Colin Allen. Moral Machines: Teaching Robots Right from Wrong. Oxford University Press, November 2008.
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Hanabi is NP-complete, Even for Cheaters who Look at Their Cards
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).
algorithmic combinatorial game theory
sorting
4:1-4:17
Regular Paper
Jean-Francois
Baffier
Jean-Francois Baffier
Man-Kwun
Chiu
Man-Kwun Chiu
Yago
Diez
Yago Diez
Matias
Korman
Matias Korman
Valia
Mitsou
Valia Mitsou
André
van Renssen
André van Renssen
Marcel
Roeloffzen
Marcel Roeloffzen
Yushi
Uno
Yushi Uno
10.4230/LIPIcs.FUN.2016.4
Antoine Bauza. Hanabi. URL: http://www.antoinebauza.fr/?tag=hanabi.
http://www.antoinebauza.fr/?tag=hanabi
BoardGameGeek. URL: https://boardgamegeek.com/boardgame/98778/hanabi.
https://boardgamegeek.com/boardgame/98778/hanabi
Alex Churchill. Magic: The gathering is Turing complete. Unpublished manuscript available at http://www.toothycat.net/~hologram/Turing/index.html.
http://www.toothycat.net/~hologram/Turing/index.html
Christopher Cox, Jessica De Silva, Philip Deorsey, Franklin H. J. Kenter, Troy Retter, and Josh Tobin. How to make the perfect fireworks display: Two strategies for Hanabi. Mathematics Magazine, 88(5):323-336, 2015.
Erik D. Demaine. Personal communication.
Erik D. Demaine. Playing games with algorithms: Algorithmic combinatorial game theory. CoRR, cs.CC/0106019v2, 2008.
Erik D. Demaine, Martin L. Demaine, Nicholas J. A. Harvey, Ryuhei Uehara, Takeaki Uno, and Yushi Uno. UNO is hard, even for a single player. Theor. Comp. Sci., 521:51-61, 2014.
Spiel des Jahres award. URL: http://www.spieldesjahres.de/en/hanabi.
http://www.spieldesjahres.de/en/hanabi
Aviezri S. Fraenkel and David Lichtenstein. Computing a perfect strategy for n x n chess requires time exponential in n. J. Comb. Theory, Ser. A, 31(2):199-214, 1981.
Martin Gardner. Mathematical Games: The Entire Collection of His Scientific American Columns. The Mathematical Association of America, 2005.
Robert Hearn and Erik D. Demaine. Games, Puzzles, and Computation. A. K. Peters, 2009.
Michael Lampis and Valia Mitsou. The computational complexity of the game of set and its theoretical applications. In 11^th Latin American Symposium, pages 24-34. Springer, 2014.
Kenichiro Nakai and Yasuhiko Takenaga. NP-completeness of pandemic. JIP, 20(3):723-726, 2012.
Hirotaka Osawa. Solving Hanabi: Estimating hands by opponent’s actions in cooperative game with incomplete information. In AAAI workshop: Computer Poker and Imperfect Information, pages 37-43, 2015.
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Selenite Towers Move Faster Than Hanoï Towers, But Still Require Exponential Time
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).
Brähma tower
Disk Pile
Hanoi Tower
Levitating Tower
Recursivity
5:1-5:20
Regular Paper
Jérémy
Barbay
Jérémy Barbay
10.4230/LIPIcs.FUN.2016.5
J.-P. Allouche and F. Dress. Tours de Hanoï et automates. RAIRO, Informatique Théorique et applications, 24(1):1-15, 1990.
M.D. Atkinson. The cyclic towers of Hanoï. Information Processing Letters (IPL), 13(118-119), 1981.
W. R. Ball. Mathematical Recreations and Essays. McMillan, London, 1892.
J. S. Frame and B. M. Stewart. Solution of problem no 3918. American Mathematics Monthly (AMM), 48:216-219, 1941.
Édouard Lucas. La tour d'Hanoï, véritable casse-tête annamite. In a puzzle game., Amiens, 1883. Jeu rapporté du Tonkin par le professeur N.Claus (De Siam).
Édouard Lucas. Récréations Mathématiques, volume II. Gauthers-Villars, Paris, quai des Augustins, 55, 1883.
D. Wood. The towers of Brahma and Hanoï revisited. Journal of Recreational Mathematics (JRM), 14(1):17-24, 1981.
Creative Commons Attribution 3.0 Unported license
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Algorithms and Insights for RaceTrack
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.
Racetrack
State-graph
complexity
6:1-6:14
Regular Paper
Michael A.
Bekos
Michael A. Bekos
Till
Bruckdorfer
Till Bruckdorfer
Henry
Förster
Henry Förster
Michael
Kaufmann
Michael Kaufmann
Simon
Poschenrieder
Simon Poschenrieder
Thomas
Stüber
Thomas Stüber
10.4230/LIPIcs.FUN.2016.6
Kristian Ahlmann-Ohlsen. Applying binary decision diagrams to solve the shortest path problem in vectorrace, 2005.
Jeff Erickson. How hard is optimal racing?, 2009. URL: http://3dpancakes.typepad.com/ernie/2009/06/how-hard-is-optimal-racing.html.
http://3dpancakes.typepad.com/ernie/2009/06/how-hard-is-optimal-racing.html
Martin Gardner. Mathematical games - Sim, chomp and race track: new games for the intellect (and not for lady luck). Scientific American, 228(1):108-115, 1973.
Peter E. Hart, Nils J. Nilsson, and Bertram Raphael. A formal basis for the heuristic determination of minimum cost paths. IEEE Trans. Systems, Science and Cybernetics, SSC-4(2):100-107, 1968.
Markus Holzer and Pierre McKenzie. The computational complexity of racetrack. In Paolo Boldi and Luisa Gargano, editors, FUN 2010, volume 6099 of LNCS, pages 260-271. Springer, 2010.
Robert Olsson and Andreas Tarandi. A genetic algorithm in the game racetrack, 2011.
Jakob Schmid. VectorRace - finding the fastest path through a two-dimensional track, 2005. URL: http://schmid.dk/articles/vectorRace.pdf.
http://schmid.dk/articles/vectorRace.pdf
Wikipedia. Racetrack (game). URL: https://en.wikipedia.org/wiki/Racetrack_game.
https://en.wikipedia.org/wiki/Racetrack_game
Creative Commons Attribution 3.0 Unported license
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Resource Optimization for Program Committee Members: A Subreview Article
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.
Scheduling
Delegation
Subreviews
7:1-7:20
Regular Paper
Michael A.
Bender
Michael A. Bender
Samuel
McCauley
Samuel McCauley
Bertrand
Simon
Bertrand Simon
Shikha
Singh
Shikha Singh
Frédéric
Vivien
Frédéric Vivien
10.4230/LIPIcs.FUN.2016.7
Dan Alistarh, Michael A. Bender, Seth Gilbert, and Rachi Guerraoui. How to allocate tasks asynchronously. In Proc. of the 53nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 331-340, 2012.
Chumki Basu, Haym Hirsh, William W. Cohen, and Craig Nevill-Manning. Recommending papers by mining the web. In Proc. of the IJCAI Workshop on Learning about Users, 1999.
Michael A. Bender and Cynthia A. Phillips. Scheduling DAGs on asynchronous processors. In Proc. of the 19th Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 35-45, 2007.
Salem Benferhat and Jérôme Lang. Conference paper assignment. International Journal of Intelligent Systems, 16(10):1183-1192, 2001.
George Bosilca, Aurélien Bouteiller, Élisabeth Brunet, Franck Cappello, Jack Dongarra, Amina Guermouche, Thomas Hérault, Yves Robert, Frédéric Vivien, and Dounia Zaidouni. Unified Model for Assessing Checkpointing Protocols at Extreme-Scale. Concurrency and Computation: Practice and Experience, 26(17):2727-2810, 2013.
Henri Casanova, Fanny Dufossé, Yves Robert, and Frédéric Vivien. Mapping applications on volatile resources. International Journal of High Performance Computing Applications, 29(1):19, 2015. URL: http://dx.doi.org/10.1177/1094342013518806.
http://dx.doi.org/10.1177/1094342013518806
Henri Casanova, Dounia Zaidouni, and Frédéric Vivien. Using replication for resilience on exascale systems. In Thomas Hérault and Yves Robert, editors, Fault-Tolerance Techniques for High-Performance Computing, page 50. Springer, 2015.
Bogdan S. Chlebus and Dariusz R. Kowalski. Cooperative asynchronous update of shared memory. In Proc. of the 37th Annual ACM Symposium on Theory of Computing (STOC), pages 733-739, 2005.
Graham Cormode. How not to review a paper: The tools and techniques of the adversarial reviewer. ACM SIGMOD Record, 37(4):100-104, 2009.
John T. Daly. A higher order estimate of the optimum checkpoint interval for restart dumps. Future Generation Computer Systems, 22(3):303-312, 2004.
Susan T. Dumais and Jakob Nielsen. Automating the assignment of submitted manuscripts to reviewers. In Proc. of the 15th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, pages 233-244, 1992.
Hiroshi Fujiwara and Kazuo Iwama. Average-case competitive analyses for ski-rental problems. Algorithmica, 42(1):95-107, 2005.
Judy Goldsmith and Robert H. Sloan. The AI conference paper assignment problem. In Proc. of the AAAI Workshop on Preference Handling for Artificial Intelligence, Vancouver, pages 53-57, 2007.
David Hartvigsen, Jerry C. Wei, and Richard Czuchlewski. The conference paper-reviewer assignment problem. Decision Sciences, 30(3):865-876, 1999.
Thomas Hérault and Yves Robert. Fault-Tolerance Techniques for High-Performance Computing. Springer, 2015. URL: http://dx.doi.org/10.1007/978-3-319-20943-2.
http://dx.doi.org/10.1007/978-3-319-20943-2
Seth Hettich and Michael J. Pazzani. Mining for proposal reviewers: lessons learned at the national science foundation. In Proc. of the 12th Annual ACM International Conference on Knowledge Discovery and Data Mining (KDD), pages 862-871, 2006.
Z. M. Kedem, K. V. Palem, M. O. Rabin, and A. Raghunathan. Efficient program transformation for resilient parallel computation via randomization. In Proc. of the 24th Annual ACM Symposium on the Theory of Computing (STOC), pages 306-317, 1992.
Zvi M. Kedem, Krishna V. Palem, and Paul G. Spirakis. Efficient robust parallel computations. In Proc. of the 22rd Annual ACM Symposium on Theory of Computing (STOC), pages 138-148, 1990.
Dariusz R. Kowalski and Alexander A. Shvartsman. Writing-all deterministically and optimally using a nontrivial number of asynchronous processors. ACM Transactions on Algorithms, 4:33:1-33:22, 2008.
Julien Lesca and Patrice Perny. LP solvable models for multiagent fair allocation problems. In Proc. of the 19th European Conference on Artificial Intelligence, pages 393-398. IOS Press, 2010.
Xinlian Li and Toyohide Watanabe. Automatic paper-to-reviewer assignment, based on the matching degree of the reviewers. Procedia Computer Science, 22:633-642, 2013.
Charles Martel and Ramesh Subramonian. On the complexity of certified write-all algorithms. Journal of Algorithms, 16:361-387, 1994.
Juan Julián Merelo-Guervós and Pedro Castillo-Valdivieso. Conference paper assignment using a combined greedy/evolutionary algorithm. In Parallel Problem Solving from Nature-PPSN VIII, pages 602-611, 2004.
David Mimno and Andrew McCallum. Expertise modeling for matching papers with reviewers. In Proc. of the 13th Annual ACM International Conference on Knowledge Discovery and Data Mining (KDD), pages 500-509, 2007.
Jaroslaw Protasiewicz. A support system for selection of reviewers. In IEEE International Conference on Systems, Man and Cybernetics (SMC), pages 3062-3065, 2014.
Fan Wang, Ben Chen, and Zhaowei Miao. A survey on reviewer assignment problem. In New frontiers in applied artificial intelligence, pages 718-727. Springer, 2008.
Fan Wang, Ning Shi, and Ben Chen. A comprehensive survey of the reviewer assignment problem. International Journal of Information Technology &Decision Making, 9(04):645-668, 2010.
David Yarowsky and Radu Florian. Taking the load off the conference chairs: towards a digital paper-routing assistant. In Proc. of the Joint SIGDAT Conference on Empirical Methods in NLP and Very-Large Corpora, 1999.
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Physical Zero-Knowledge Proofs for Akari, Takuzu, Kakuro and KenKen
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.
Physical Cryptography
ZKP
Games
Akari
Kakuro
KenKen
Takuzu
8:1-8:20
Regular Paper
Xavier
Bultel
Xavier Bultel
Jannik
Dreier
Jannik Dreier
Jean-Guillaume
Dumas
Jean-Guillaume Dumas
Pascal
Lafourcade
Pascal Lafourcade
10.4230/LIPIcs.FUN.2016.8
Michael Ben-Or, Oded Goldreich, Shafi Goldwasser, Johan Håstad, Joe Kilian, Silvio Micali, and Phillipe Rogaway. Everything provable is provable in zero-knowledge. In CRYPTO'88, pages 37-56. Springer, 1990.
Marzio De Biasi. Binary puzzle is NP-complete. 2012. URL: http://www.nearly42.org/vdisk/cstheory/binaryp.pdf.
http://www.nearly42.org/vdisk/cstheory/binaryp.pdf
Manuel Blum, Paul Feldman, and Silvio Micali. Non-interactive zero-knowledge and its applications. In STOC'88, pages 103-112. ACM, 1988.
Yu-Feng Chien and Wing-Kai Hon. Cryptographic and physical zero-knowledge proof: From sudoku to nonogram. In Fun with Algorithms, 5th International Conference, FUN'10, volume 6099 of LNCS, pages 102-112, 2010.
Erik D. Demaine. Playing games with algorithms: Algorithmic combinatorial game theory. In Mathematical Foundations of Computer Science 2001, volume 2136 of LNCS, pages 18-32, 2001.
Jannik Dreier, Hugo Jonker, and Pascal Lafourcade. Secure auctions without cryptography. In Fun with Algorithms, 7th International Conference, FUN'14, pages 158-170, 2014.
Oded Goldreich and Ariel Kahan. How to construct constant-round zero-knowledge proof systems for NP. Journal of Cryptology, 9(3):167-189, 1991.
Shafi Goldwasser, Silvio Micali, and Charles Rackoff. The knowledge complexity of interactive proof-systems. In STOC'85, pages 291-304. ACM, 1985.
Ronen Gradwohl, Moni Naor, Benny Pinkas, and Guy N. Rothblum. Cryptographic and physical zero-knowledge proof systems for solutions of sudoku puzzles. In Fun with Algorithms, 4th International Conference, FUN'07, pages 166-182. Springer-Verlag, 2007.
Kazuya Haraguchi and Hirotaka Ono. BLOCKSUM is NP-complete. IEICE Transactions, 96-D(3):481-488, 2013.
Kazuya Haraguchi and Hirotaka Ono. How simple algorithms can solve latin square completion-type puzzles approximately. JIP, 23(3):276-283, 2015.
Graham Kendall, Andrew J. Parkes, and Kristian Spoerer. A survey of NP-complete puzzles. ICGA Journal, 31(1):13-34, 2008.
Jonas Kölker. I/O-Efficient Multiparty Computation, Formulaic Secret Sharing and NP-Complete Puzzles. PhD thesis, Aarhus University, 2012.
Brandon McPhail. Light up is NP-complete. feb 2005. URL: http://www.mountainvistasoft.com/docs/lightup-is-np-complete.pdf.
http://www.mountainvistasoft.com/docs/lightup-is-np-complete.pdf
Alfred J. Menezes, Scott A. Vanstone, and Paul C. Van Oorschot. Handbook of Applied Cryptography. CRC Press, Inc., Boca Raton, FL, USA, 1st edition, 1996.
Takaaki Mizuki and Hiroki Shizuya. Practical card-based cryptography. In Fun with Algorithms, 7th International Conference, FUN'14, pages 313-324, 2014.
Moni Naor, Yael Naor, and Omer Reingold. Applied kid cryptography or how to convince your children you are not cheating. In Eurocrypt’94, pages 1-12, 1999.
Jean-Jacques Quisquater, Myriam Quisquater, Muriel Quisquater, Michaël Quisquater, Louis Guillou, Marie Annick Guillou, Gaïd Guillou, Anna Guillou, Gwenolé Guillou, Soazig Guillou, and Thomas A. Berson. How to explain zero-knowledge protocols to your children. In CRYPTO'89, pages 628-631. Springer-Verlag, 1990.
J. Barkley Rosser and Lowell Schoenfeld. Approximate formulas for some functions of prime numbers. Illinois Journal of Mathematics, 6:64-94, 1962.
Dennis Shasha. Upstart puzzles: Proving without teaching/teaching without proving. Commun. ACM, 57(11):120-120, 2014.
Will Shortz. Easy Kakuro: 100 Addictive Logic Puzzles. St. Martin’s Griffin, 2006.
Seta Takahiro. The complexities of puzzles, cross sum and their another solution problems(asp). Master’s thesis, University of Tokyo, 2001.
Putranto Hadi Utomo and Ruud Pellikaan. Binary puzzles as an erasure decoding problem. In Proceedings of the 36th WIC Symposium on Information Theory in the Benelux, pages 129-134, 2015.
Creative Commons Attribution 3.0 Unported license
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Analyzing and Comparing On-Line News Sources via (Two-Layer) Incremental Clustering
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.
text mining
incremental clustering
on-line news
9:1-9:14
Regular Paper
Francesco
Cambi
Francesco Cambi
Pierluigi
Crescenzi
Pierluigi Crescenzi
Linda
Pagli
Linda Pagli
10.4230/LIPIcs.FUN.2016.9
J. Azzopardi and C. Staff. Incremental Clustering of News Reports. Algorithms, 5:364-378, 2012.
D. Bhattacharya and S. Ram. Sharing News Articles Using 140 Characters: A Diffusion Analysis on Twitter. In IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, pages 966-971, 2012.
Jon Borglund. Event-Centric Clustering of News Articles. Technical report, Department of Information Technology, University of Uppsala, 2013.
T.F. Cox and M.A.A. Cox. Multidimensional Scaling (2nd ed.). Chapman and Hall, 2000.
S. Edunov, C.G. Diuk, I.O. Filiz, S. Bhagat, and M. Burke. Three and a half degrees of separation, 2016. URL: http://research.facebook.com/blog/.
http://research.facebook.com/blog/
R. Fagin, R. Kumar, and D. Sivakumar. Comparing Top K Lists. In Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 28-36, 2003.
M. Kendall and J. D. Gibbons. Rank Correlation Methods. Edward Arnold, 1990.
J. Leskovec, A. Rajaraman, and J.D. Ullman. Mining of Massive Datasets. Cambridge University Press, 2014.
Vladimir I. Levenshtein. Binary codes capable of correcting deletions, insertions, and reversals. Soviet Physics Doklady, 10:707-710, 1966.
J.B. Lovins. Development of a Stemming Algorithm. Mechanical Translation and Computational Linguistics, 11:22-31, 1968.
Parse.ly. What is the Lifespan of an Article?, 2015. URL: http://parsely.com.
http://parsely.com
G. Petkos, S. Papadopoulos, and Y. Kompatsiaris. Two-level Message Clustering for Topic Detection in Twitter. In SNOW 2014 Data Challenge co-located with 23rd International World Wide Web Conference, pages 49-56, 2014.
Wikipedia - News Agency. URL: https://en.wikipedia.org/wiki/News_agency.
https://en.wikipedia.org/wiki/News_agency
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Spy-Game on Graphs
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.
graph
two-player games
cops and robber games
complexity
10:1-10:16
Regular Paper
Nathann
Cohen
Nathann Cohen
Mathieu
Hilaire
Mathieu Hilaire
Nícolas A.
Martins
Nícolas A. Martins
Nicolas
Nisse
Nicolas Nisse
Stéphane
Pérennes
Stéphane Pérennes
10.4230/LIPIcs.FUN.2016.10
M. Aigner and M. Fromme. A game of cops and robbers. Discrete Applied Mathematics, 8:1-12, 1984.
N. Alon and A. Mehrabian. On a generalization of Meyniel’s conjecture on the cops and robbers game. Electr. J. Comb., 18(1), 2011.
A. Bonato, E. Chiniforooshan, and P. Pralat. Cops and robbers from a distance. Theor. Comput. Sci., 411(43):3834-3844, 2010.
A. Bonato and R. Nowakovski. The game of Cops and Robber on Graphs. American Math. Soc., 2011.
J. Chalopin, V. Chepoi, N. Nisse, and Y. Vaxès. Cop and robber games when the robber can hide and ride. SIAM J. Discrete Math., 25(1):333-359, 2011.
Uriel Feige. A threshold of log n for approximating set cover. J. ACM, 45(4):634-652, 1998.
F. V. Fomin, P. A. Golovach, J. Kratochvíl, N. Nisse, and K. Suchan. Pursuing a fast robber on a graph. Theor. Comput. Sci., 411(7-9):1167-1181, 2010.
F. V. Fomin, P. A. Golovach, and D. Lokshtanov. Cops and robber game without recharging. In 12th Scandinavian Symp. and Workshops on Algorithm Theory (SWAT), volume 6139 of LNCS, pages 273-284. Springer, 2010.
F.V. Fomin, P. A. Golovach, and P. Pralat. Cops and robber with constraints. SIAM J. Discrete Math., 26(2):571-590, 2012.
W. Goddard, S.M. Hedetniemi, and S.T. Hedetniemi. Eternal security in graphs. J. Combin.Math.Combin.Comput., 52, 2005.
John L. Goldwasser and William Klostermeyer. Tight bounds for eternal dominating sets in graphs. Discrete Mathematics, 308(12):2589-2593, 2008.
William B. Kinnersley. Cops and robbers is exptime-complete. J. Comb. Theory, Ser. B, 111:201-220, 2015.
W.F. Klostermeyer and G MacGillivray. Eternal dominating sets in graphs. J. Combin.Math.Combin.Comput., 68, 2009.
W.F. Klostermeyer and C.M. Mynhardt. Graphs with equal eternal vertex cover and eternal domination numbers. Discrete Mathematics, 311(14):1371-1379, 2011.
R. J. Nowakowski and P. Winkler. Vertex-to-vertex pursuit in a graph. Discrete Maths, 43:235-239, 1983.
Creative Commons Attribution 3.0 Unported license
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The Complexity of Snake
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.
Games
Puzzles
Motion Planning
Nondeterministic Constraint Logic
PSPACE
11:1-11:13
Regular Paper
Marzio
De Biasi
Marzio De Biasi
Tim
Ophelders
Tim Ophelders
10.4230/LIPIcs.FUN.2016.11
Elias Dahlhaus, Peter Horák, Mirka Miller, and Joseph F. Ryan. The train marshalling problem. Discrete Applied Mathematics, 103(1-3):41-54, 2000. URL: http://dx.doi.org/10.1016/S0166-218X(99)00219-X.
http://dx.doi.org/10.1016/S0166-218X(99)00219-X
Erik D. Demaine and Robert A. Hearn. Playing games with algorithms: Algorithmic combinatorial game theory. In Games of No Chance 3, volume 56 of Mathematical Sciences Research Institute Publications, pages 3-56. Cambridge University Press, 2009.
Robert A Hearn and Erik D Demaine. PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theoretical Computer Science, 343(1):72-96, 2005.
Robert A. Hearn and Erik D. Demaine. Games, puzzles and computation. A K Peters, 2009.
Graham Kendall, Andrew J. Parkes, and Kristian Spoerer. A survey of NP-complete puzzles. ICGA Journal, 31(1):13-34, 2008.
Irina Kostitsyna and Valentin Polishchuk. Simple wriggling is hard unless you are a fat hippo. Theory of Computing Systems, 50(1):93-110, 2012. URL: http://dx.doi.org/10.1007/s00224-011-9337-4.
http://dx.doi.org/10.1007/s00224-011-9337-4
J.-C. Latombe. Robot Motion Planning. Kluwer, Boston, MA, 1991.
Amit Pamecha, Imme Ebert-Uphoff, and Gregory S. Chirikjian. Useful metrics for modular robot motion planning. IEEE Transactions on Robotics and Automation, 13(4):531-545, 1997. URL: http://dx.doi.org/10.1109/70.611311.
http://dx.doi.org/10.1109/70.611311
Christos H. Papadimitriou and Umesh V. Vazirani. On two geometric problems related to the traveling salesman problem. Journal of Algorithms, 5(2):231-246, 1984. URL: http://dx.doi.org/10.1016/0196-6774(84)90029-4.
http://dx.doi.org/10.1016/0196-6774(84)90029-4
Roberto Tamassia. On embedding a graph in the grid with the minimum number of bends. SIAM Journal on Computing, 16(3):421-444, 1987. URL: http://dx.doi.org/10.1137/0216030.
http://dx.doi.org/10.1137/0216030
Christopher Umans and William Lenhart. Hamiltonian cycles in solid grid graphs. In FOCS, pages 496-505, 1997. URL: http://dx.doi.org/10.1109/SFCS.1997.646138.
http://dx.doi.org/10.1109/SFCS.1997.646138
Tom C. van der Zanden. Parameterized Complexity of Graph Constraint Logic. In IPEC, pages 282-293, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.IPEC.2015.282.
http://dx.doi.org/10.4230/LIPIcs.IPEC.2015.282
Giovanni Viglietta. Gaming is a hard job, but someone has to do it! Theory of Computing Systems, 54(4):595-621, 2014. URL: http://dx.doi.org/10.1007/s00224-013-9497-5.
http://dx.doi.org/10.1007/s00224-013-9497-5
Zhi Yan, Nicolas Jouandeau, and Arab Ali Cherif. A survey and analysis of multi-robot coordination. International Journal of Advanced Robotic Systems, 10:399, 2013.
Creative Commons Attribution 3.0 Unported license
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The Fewest Clues Problem
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.
computational complexity
pencil-and-paper puzzles
hardness reductions
12:1-12:12
Regular Paper
Erik D.
Demaine
Erik D. Demaine
Fermi
Ma
Fermi Ma
Ariel
Schvartzman
Ariel Schvartzman
Erik
Waingarten
Erik Waingarten
Scott
Aaronson
Scott Aaronson
10.4230/LIPIcs.FUN.2016.12
Charles J Colbourn. The complexity of completing partial Latin squares. Discrete Applied Mathematics, 8(1):25-30, 1984.
Erik D Demaine, Yoshio Okamoto, Ryuhei Uehara, and Uno Yushi. Computational complexity and an integer programming model of shakashaka. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 97(6):1213-1219, 2014.
Ian Holyer. The NP-completeness of some edge-partition problems. SIAM Journal on Computing, 10(4):713-717, 1981.
David Lichtenstein. Planar formulae and their uses. SIAM Journal on Computing, 11(2):329-343, 1982. URL: http://dx.doi.org/10.1137/0211025.
http://dx.doi.org/10.1137/0211025
Brandon McPhail. Light up is NP-complete. Unpublished manuscript, 2005.
Wolfgang Mulzer and Günter Rote. Minimum-weight triangulation is NP-hard. CoRR, abs/cs/0601002, 2006. URL: http://arxiv.org/abs/cs/0601002.
http://arxiv.org/abs/cs/0601002
Takahiro Seta. The complexities of puzzles, cross sum and their another solution problems (ASP). Senior thesis, Univ. of Tokyo, Dept. of Information Science, Tokyo, Japan, Feb 2001. URL: http://www-imai.is.s.u-tokyo.ac.jp/~seta/paper/senior_thesis/seniorthesis.ps.
http://www-imai.is.s.u-tokyo.ac.jp/~seta/paper/senior_thesis/seniorthesis.ps
Yato Takayuki and Seta Takahiro. Complexity and completeness of finding another solution and its application to puzzles. IEICE transactions on fundamentals of electronics, communications and computer sciences, 86(5):1052-1060, 2003.
Seinosuke Toda. PP is as hard as the polynomial-time hierarchy. SIAM J. Comput., 20(5):865-877, October 1991. URL: http://dx.doi.org/10.1137/0220053.
http://dx.doi.org/10.1137/0220053
Nobuhisa Ueda and Tadaaki Nagao. NP-completeness results for nonogram via parsimonious reductions. preprint, 1996.
Leslie G. Valiant. The complexity of computing the permanent. Theoretical Computer Science, 8(2):189-201, 1979. URL: http://dx.doi.org/10.1016/0304-3975(79)90044-6.
http://dx.doi.org/10.1016/0304-3975(79)90044-6
Leslie G. Valiant. The complexity of enumeration and reliability problems. SIAM Journal on Computing, 8(3):410-421, 1979. URL: http://dx.doi.org/10.1137/0208032.
http://dx.doi.org/10.1137/0208032
Leslie G. Valiant and Vijay V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47(C):85-93, 1986. URL: http://dx.doi.org/10.1016/0304-3975(86)90135-0.
http://dx.doi.org/10.1016/0304-3975(86)90135-0
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Super Mario Bros. is Harder/Easier Than We Thought
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.
video games
computational complexity
PSPACE
13:1-13:14
Regular Paper
Erik D.
Demaine
Erik D. Demaine
Giovanni
Viglietta
Giovanni Viglietta
Aaron
Williams
Aaron Williams
10.4230/LIPIcs.FUN.2016.13
Greg Aloupis, Erik D. Demaine, Alan Guo, and Giovanni Viglietta. Classic Nintendo games are (computationally) hard. Theoretical Computer Science, 586:135-160, 2015.
Speed Demos Archive. Super Mario Bros. URL: http://kb.speeddemosarchive.com/Super_Mario_Bros.
http://kb.speeddemosarchive.com/Super_Mario_Bros.
doppleganger. A comprehensive Super Mario Bros. disassembly. URL: http://giovanniviglietta.com/files/SMB/source.asm.
http://giovanniviglietta.com/files/SMB/source.asm
Michael Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, 1979.
TASVideos. Super Mario Bros. URL: http://tasvideos.org/GameResources/NES/SuperMarioBros.html.
http://tasvideos.org/GameResources/NES/SuperMarioBros.html
TV Tropes. Nintendo Hard. URL: http://tvtropes.org/pmwiki/pmwiki.php/Main/NintendoHard.
http://tvtropes.org/pmwiki/pmwiki.php/Main/NintendoHard
Tom C. van der Zanden. Parameterized complexity of graph constraint logic. In Proceedings of the 10th International Symposium on Parameterized and Exact Computation (IPEC 2015), Leibniz International Proceedings in Informatics (LIPIcs), pages 282-293. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2015. URL: http://dx.doi.org/10.4230/LIPIcs.IPEC.2015.282.
http://dx.doi.org/10.4230/LIPIcs.IPEC.2015.282
Vargomax V. Vargomax. Generalized Super Mario Bros. is NP-complete. In Proceedings of the 6th Binarennial Workshop about Symposium on Robot Dance Party of Conference in Celebration of Harry Q. Bovik’s 0x40th Birthday (SIGBOVIK 2007), pages 87-88, 2007.
Giovanni Viglietta. Gaming is a hard job, but someone has to do it! Theory of Computing Systems, 54(4):595-621, 2014.
Giovanni Viglietta. Lemmings is PSPACE-complete. Theoretical Computer Science, 586:120-134, 2015.
Video Game Sales Wiki. Mario. URL: http://vgsales.wikia.com/wiki/Mario.
http://vgsales.wikia.com/wiki/Mario
Creative Commons Attribution 3.0 Unported license
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A Rupestrian Algorithm
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.
mobile finite-state machines
self-healing distributed algorithms
14:1-14:20
Regular Paper
Giuseppe A.
Di Luna
Giuseppe A. Di Luna
Paola
Flocchini
Paola Flocchini
Giuseppe
Prencipe
Giuseppe Prencipe
Nicola
Santoro
Nicola Santoro
Giovanni
Viglietta
Giovanni Viglietta
10.4230/LIPIcs.FUN.2016.14
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Building a Better Mouse Maze
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.
graph
evolutionary
genetic algorithm
traversal
15:1-15:12
Regular Paper
Jessica
Enright
Jessica Enright
John D.
Faben
John D. Faben
10.4230/LIPIcs.FUN.2016.15
Bilgehan Çavdaroğlu and Fuat Balcı. Mice can count and optimize count-based decisions. Psychonomic Bulletin &Review, pages 1-6, 2015. URL: http://dx.doi.org/10.3758/s13423-015-0957-6.
http://dx.doi.org/10.3758/s13423-015-0957-6
Andrew Drizen. private communication, 2016.
Jane Green, editor. Personal Computer World: Best of PCW Software for the Commodore 64. Century Communications, 1984.
Zbigniew Michalewicz and David B. Fogel. How to Solve It: Modern Heuristics. Springer, 2000.
Creative Commons Attribution 3.0 Unported license
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Recognizing a DOG is Hard, But Not When It is Thin and Unit
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.
graph drawing
planar graphs
disk intersection graphs
16:1-16:12
Regular Paper
William
Evans
William Evans
Mereke
van Garderen
Mereke van Garderen
Maarten
Löffler
Maarten Löffler
Valentin
Polishchuk
Valentin Polishchuk
10.4230/LIPIcs.FUN.2016.16
Patrizio Angelini, Giordano Da Lozzo, Marco Di Bartolomeo, Giuseppe Di Battista, Seok-Hee Hong, Maurizio Patrignani, and Vincenzo Roselli. Anchored drawings of planar graphs. In Graph Drawing, volume 8871 of Lecture Notes in Computer Science, pages 404-415. Springer, 2014.
Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, and Fabrizio Frati. Strip planarity testing. In Stephen K. Wismath and Alexander Wolff, editors, Graph Drawing, volume 8242 of Lecture Notes in Computer Science, pages 37-48. Springer, 2013.
Esther M Arkin, Michael A Bender, Erik D Demaine, Sándor P Fekete, Joseph SB Mitchell, and Saurabh Sethia. Optimal covering tours with turn costs. SIAM Journal on Computing, 35(3):531-566, 2005.
Esther M Arkin, Sándor P Fekete, Kamrul Islam, Henk Meijer, Joseph SB Mitchell, Yurai Núñez-Rodríguez, Valentin Polishchuk, David Rappaport, and Henry Xiao. Not being (super) thin or solid is hard: A study of grid Hamiltonicity. Computational Geometry, 42(6):582-605, 2009.
Lali Barrière, Pierre Fraigniaud, Lata Narayanan, and Jaroslav Opatrny. Robust position-based routing in wireless ad hoc networks with irregular transmission ranges. WCMC, 3(2):141-153, 2003. URL: http://dx.doi.org/10.1002/wcm.108.
http://dx.doi.org/10.1002/wcm.108
Prosenjit Bose, Pat Morin, Ivan Stojmenović, and Jorge Urrutia. Routing with guaranteed delivery in ad hoc wireless networks. Wireless networks, 7(6):609-616, 2001.
Kevin Buchin, Irina Kostitsyna, Maarten Löffler, and Rodrigo I Silveira. Region-based approximation algorithms for visibility between imprecise locations. In ALENEX, pages 94-103. SIAM, 2015.
Kevin Buchin, Maarten Löffler, Pat Morin, and Wolfgang Mulzer. Preprocessing imprecise points for Delaunay triangulation: Simplified and extended. Algorithmica, 61(3):674-693, 2011. URL: http://dx.doi.org/10.1007/s00453-010-9430-0.
http://dx.doi.org/10.1007/s00453-010-9430-0
Hubie Chen. A rendezvous of logic, complexity, and algebra. ACM Comput. Surv., 42(1):2:1-2:32, December 2009. URL: http://dx.doi.org/10.1145/1592451.1592453.
http://dx.doi.org/10.1145/1592451.1592453
M.B. Cozzens. Higher and Multi-dimensional Analogues of Interval Graphs. PhD thesis, Rutgers University, 1981.
Hongwei Du, Xiaohua Jia, Deying Li, and Weili Wu. Coloring of double disk graphs. J. Global Opt, 28(1):115-119, 2004. URL: http://dx.doi.org/10.1023/B:JOGO.0000006750.85332.0f.
http://dx.doi.org/10.1023/B:JOGO.0000006750.85332.0f
Alon Efrat, Sándor P Fekete, Joseph SB Mitchell, Valentin Polishchuk, and Jukka Suomela. Improved approximation algorithms for relay placement. ACM Transactions on Algorithms, 12(2):20, 2015.
M. Gromov. Hyperbolic groups. In S. M. Gersten, editor, Essays in Group Theory, pages 75-263. Springer New York, 1987.
Marja Hassinen, Joel Kaasinen, Evangelos Kranakis, Valentin Polishchuk, Jukka Suomela, and Andreas Wiese. Analysing local algorithms in location-aware quasi-unit-disk graphs. Discr Appl Math, 159(15):1566-1580, 2011. URL: http://dx.doi.org/10.1016/j.dam.2011.05.004.
http://dx.doi.org/10.1016/j.dam.2011.05.004
Jean-Claude Hausmann. On the Vietoris-Rips complexes and a cohomology theory for metric spaces. In F. Quinn, editor, Annals of Mathematics Studies, volume 138, pages 175-188, Princeton, N.J., 1995. Princeton University Press. Prospects in topology : proceedings of a conference in honor of William Browder.
Hiroshi Imai and Takao Asano. Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. Journal of Algorithms, 4(4):310-323, 1983. URL: http://dx.doi.org/http://dx.doi.org/10.1016/0196-6774(83)90012-3.
http://dx.doi.org/http://dx.doi.org/10.1016/0196-6774(83)90012-3
Balázs Keszegh, János Pach, and Domotor Palvolgyi. Drawing planar graphs of bounded degree with few slopes. SIAM Journal on Discrete Mathematics, 27(2):1171-1183, 2013.
Alexander Kröller, Sándor P Fekete, Dennis Pfisterer, and Stefan Fischer. Deterministic boundary recognition and topology extraction for large sensor networks. In SoDA, pages 1000-1009, 2006.
Sven O Krumke, Madhav V Marathe, and SS Ravi. Models and approximation algorithms for channel assignment in radio networks. Wireless networks, 7(6):575-584, 2001.
Fabian Kuhn, Thomas Moscibroda, and Rogert Wattenhofer. Unit disk graph approximation. In Proceedings of the 2004 joint workshop on Foundations of mobile computing, pages 17-23. ACM, 2004.
Fabian Kuhn, Roger Wattenhofer, and Aaron Zollinger. Ad hoc networks beyond unit disk graphs. Wireless Networks, 14(5):715-729, 2007. URL: http://dx.doi.org/10.1007/s11276-007-0045-6.
http://dx.doi.org/10.1007/s11276-007-0045-6
Fabian Kuhn, Rogert Wattenhofer, Yan Zhang, and Aaron Zollinger. Geometric ad-hoc routing: of theory and practice. In PoDC, pages 63-72, 2003.
David Lichtenstein. Planar formulae and their uses. SIAM J. Comput., 11(2):329-343, 1982. URL: http://dx.doi.org/10.1137/0211025.
http://dx.doi.org/10.1137/0211025
Maarten Löffler. Existence and computation of tours through imprecise points. IJCGA, 21(1):1-24, 2011. URL: http://dx.doi.org/10.1142/S0218195911003524.
http://dx.doi.org/10.1142/S0218195911003524
Maarten Löffler and Wolfgang Mulzer. Unions of onions: Preprocessing imprecise points for fast onion decomposition. JoCG, 5(1):1-13, 2014. URL: http://jocg.org/index.php/jocg/article/view/140.
http://jocg.org/index.php/jocg/article/view/140
Maarten Löffler and Marc van Kreveld. Largest and smallest convex hulls for imprecise points. Algorithmica, 56(2):235-269, 2010.
John Nolan. Bisectored unit disk graphs. Networks, 43(3):141-152, 2004. URL: http://dx.doi.org/10.1002/net.10111.
http://dx.doi.org/10.1002/net.10111
Maurizio Patrignani. On extending a partial straight-line drawing. Int. J. Found. CS, 17(5):1061-1070, 2006. URL: http://dx.doi.org/10.1142/S0129054106004261.
http://dx.doi.org/10.1142/S0129054106004261
Thomas J. Schaefer. The complexity of satisfiability problems. In SToC'78, STOC'78, pages 216-226. ACM, 1978. URL: http://dx.doi.org/10.1145/800133.804350.
http://dx.doi.org/10.1145/800133.804350
L. Vietoris. Über den höheren zusammenhang kompakter räume und eine klasse von zusammenhangstreuen abbildungen. Mathematische Annalen, 97(1):454-472, 1927. URL: http://dx.doi.org/10.1007/BF01447877.
http://dx.doi.org/10.1007/BF01447877
Yue Wang, Jie Gao, and Joseph S B Mitchell. Boundary recognition in sensor networks by topological methods. In 12th annual conference on Mobile computing and networking, pages 122-133, 2006.
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Counting Circles Without Computing Them
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.
lines arrangement
triangle
circumcircle
inscribed angle theorem
17:1-17:7
Regular Paper
Rudolf
Fleischer
Rudolf Fleischer
10.4230/LIPIcs.FUN.2016.17
ACM-ICPC International Collegiate Programming Contest. URL: https://icpc.baylor.edu.
https://icpc.baylor.edu
Problem 603D-36: Ruminations on Ruminants. Codeforces, Contest 342, Division 1, 2015, Dec 1. URL: http://codeforces.com/contest/603/problem/D.
http://codeforces.com/contest/603/problem/D
Wikipedia contributors. Inscribed angle, Date retrieved: 20 February 2016 15:40 UTC. Permanent link: URL: https://en.wikipedia.org/w/index.php?title=Inscribed_angle&oldid=699778165.
https://en.wikipedia.org/w/index.php?title=Inscribed_angle&oldid=699778165
Wikipedia contributors. Arrangement of lines, Date retrieved: 29 February 2016 15:24 UTC. Permanent link: URL: https://en.wikipedia.org/w/index.php?title=Arrangement_of_lines&oldid=702141041.
https://en.wikipedia.org/w/index.php?title=Arrangement_of_lines&oldid=702141041
R. Fleischer. FUN with implementing algorithms. In E. Lodi, L. Pagli, and N. Santoro, editors, Proceedings of the 1998 International Conference FUN with Algorithms (FUN'98), pages 88-98. Carleton Scientific, Proceedings in Informatics 4, 1999.
R. Fleischer. Problem 603D-36: Submission 14741117. Codeforces, Contest 342, Division 1, 2015, Dec 10. URL: http://codeforces.com/contest/603/submission/14741117.
http://codeforces.com/contest/603/submission/14741117
R. Fleischer. Problem 603D-36: Submission 16228941. Codeforces, Contest 342, Division 1, 2016, Jan 20. URL: http://codeforces.com/contest/603/submission/16228941.
http://codeforces.com/contest/603/submission/16228941
R. Fleischer. Problem 603D-36: Submission 16231449. Codeforces, Contest 342, Division 1, 2016, Jan 20. URL: http://codeforces.com/contest/603/submission/16231449.
http://codeforces.com/contest/603/submission/16231449
R. Fleischer and Y. Wang. On the camera placement problem. In Proceedings of the 20th International Symposium on Algorithms and Computation (ISAAC'09). Springer Lecture Notes in Computer Science 5878, pages 255-264, 2009.
M. Mirzayanov. Codeforces: The only programming contests web 2.0 platform, 2010-2016. URL: http://codeforces.com.
http://codeforces.com
Z. Shi. Problem 603D-36: Submission 15094164. Codeforces, Contest 342, Division 1, 2015, Dec 30. URL: http://codeforces.com/contest/603/submission/15094164.
http://codeforces.com/contest/603/submission/15094164
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Large Peg-Army Maneuvers
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).
Complexity of Games
Solitaire Army
18:1-18:15
Regular Paper
Luciano
Gualà
Luciano Gualà
Stefano
Leucci
Stefano Leucci
Emanuele
Natale
Emanuele Natale
Roberto
Tauraso
Roberto Tauraso
10.4230/LIPIcs.FUN.2016.18
M. Aigner. Moving into the Desert with Fibonacci. Math. Magazine, 70(1):11-21, 1997.
G. Aloupis, E. D. Demaine, A. Guo, and G. Viglietta. Classic Nintendo games are (computationally) hard. Theoretical Computer Science, 586:135-160, 2015.
J. D. Beasley. The Ins and Outs of Peg Solitaire. Oxford University Press, 1985.
J. D. Beasley. Solitaire: Recent Developments. arXiv:0811.0851, 2008.
J. D. Beasley. John and Sue Beasley’s Webpage on Peg Solitaire, 2015. URL: http://jsbeasley.co.uk/pegsol.htm.
http://jsbeasley.co.uk/pegsol.htm
G. I. Bell. A Fresh Look at Peg Solitaire. Mathe.Magazine, 80(1):16-28, 2007.
G. I. Bell, D. S. Hirschberg, and P. Guerrero-Garcia. The minimum size required of a solitaire army. arXiv/0612612, 2006.
E. R. Berlekamp, J. H. Conway, and R. K. Guy. Winning Ways for Your Mathematical Plays, Volume 2. AK Peters, 2002.
A. Bialostocki. An application of elementary group theory to central solitaire. The College Mathematics Journal, 29(3):208, 1998.
F. Chung, R. Graham, J. Morrison, and A. Odlyzko. Pebbling a Chessboard. The American Mathematical Monthly, 102(2):113-123, 1995.
B. Csakany and R. Juhasz. The Solitaire Army Reinspected. Math. Magazine, 73(5):354-362, 2000.
E. W. Dijkstra. The checkers problem told to me by M.O. Rabin, 1992. URL: http://www.cs.utexas.edu/users/EWD/ewd11xx/EWD1134.PDF.
http://www.cs.utexas.edu/users/EWD/ewd11xx/EWD1134.PDF
E. W. Dijkstra. Only a matter of style?, 1995. URL: http://www.cs.utexas.edu/users/EWD/ewd12xx/EWD1200.PDF.
http://www.cs.utexas.edu/users/EWD/ewd12xx/EWD1200.PDF
M. Gardner. The Unexpected Hanging and Other Mathematical Diversions. University of Chicago Press, 1991.
L. Guala, S. Leucci, and E. Natale. Bejeweled, Candy Crush and other match-three games are (NP-)hard. In CIG 2014, pages 1-8, 2014.
R. A. Hearn and E. D. Demaine. Games, puzzles, and computation. AK Peters, 2009.
R. Honsberger. A problem in checker jumping. Mathematical Gems II, pages 23-28, 1976.
C. Jefferson, A. Miguel, I. Miguel, and S. A. Tarim. Modelling and solving English Peg Solitaire. Computers &Operations Research, 33(10):2935-2959, 2006.
M. Kiyomi and T. Matsui. Integer Programming Based Algorithms for Peg Solitaire Problems. In Computers and Games, number 2063 in LNCS, pages 229-240. Springer, 2000.
C. Moore and D. Eppstein. One-Dimensional Peg Solitaire, and Duotaire. arXiv/0008172, 2000.
B. Ravikvmar. Peg-solitaire, string rewriting systems and finite automata. In Algorithms and Computation, volume 1350, pages 233-242. Springer, 1997.
R. Uehara and S. Iwata. Generalized Hi-Q is NP-Complete. IEICE TRANSACTIONS, E73-E(2):270-273, 1990.
Creative Commons Attribution 3.0 Unported license
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Loopless Gray Code Enumeration and the Tower of Bucharest
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.
Tower of Hanoi
Gray code
enumeration
loopless generation
19:1-19:19
Regular Paper
Felix
Herter
Felix Herter
Günter
Rote
Günter Rote
10.4230/LIPIcs.FUN.2016.19
James R. Bitner, Gideon Ehrlich, and Edward M. Reingold. Efficient generation of the binary reflected Gray code and its applications. Commun. ACM, 19(9):517-521, 1976.
Peter Buneman and Leon Levy. The towers of Hanoi problem. Information Processing Letters, 10(4-5):243-244, 1980.
Gideon Ehrlich. Loopless algorithms for generating permutations, combinations, and other combinatorial configurations. J. Assoc. Comput. Mach., 20(3):500-513, July 1973.
Martin Gardner. The curious properties of the Gray code and how it can be used to solve puzzles. Sci. American, 227:106-109, 1972.
Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. Concrete Mathematics. Addison-Wesley, 1989.
Dah-Jyh Guan. Generalized Gray codes with applications. Proc. Natl. Sci. Council, Republic of China (A), 22(6):841-848, 1998.
Leo J. Guibas, Edward M. McCreight, Michael F. Plass, and Janet R. Roberts. A new representation for linear lists. In Proceedings of the Ninth Annual ACM Symposium on Theory of Computing, STOC'77, pages 49-60, New York, NY, USA, 1977. ACM.
Felix Herter and Günter Rote. Loopless Gray code enumeration and the Tower of Bucharest. Preprint arXiv 1604.06707 [cs.DM], April 2016.
Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović, and Ciril Petr. The Tower of Hanoi - Myths and Maths. Birkhäuser, 2013.
Donald E. Knuth. Combinatorial Algorithms, Part 1, volume 4A of The Art of Computer Programming. Addison-Wesley, 2011.
Jayadev Misra. Remark on Algorithm 246. ACM Trans. Math. Software, 1(3):285, 1975.
Amir Sapir. The towers of Hanoi with forbidden moves. The Computer Journal, 47(1):20-24, 2004.
R. S. Scorer, P. M. Grundy, and C. A. B. Smith. Some binary games. The Mathematical Gazette, 28(280):96-103, 1944.
Manuel Wettstein. Counting and enumerating crossing-free geometric graphs. Preprint arXiv 1604.05350 [cs.CG], April 2016.
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Convex Configurations on Nana-kin-san Puzzle
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.
silhouette puzzles
nana-kin-san puzzle
enumeration algorithm
convex polygon
20:1-20:14
Regular Paper
Takashi
Horiyama
Takashi Horiyama
Ryuhei
Uehara
Ryuhei Uehara
Haruo
Hosoya
Haruo Hosoya
10.4230/LIPIcs.FUN.2016.20
Henry Ernest Dudeney. The Canterbury Puzzles. Dover, 1958.
Eli Fox-Epstein, Kazuho Katsumata, and Ryuhei Uehara. The Convex Configurations of "Sei Shonagon Chie no Ita," Tangram, and Other Silhouette Puzzles with Seven Pieces. IEICE Trans. on Inf. and Sys., accepted, 2016.
Martin Gardner. Origami, Eleusis, and the Soma Cube. The New Martin Gardner Mathmatical Library. Cambridge, 2008.
Jason S. Ku, Erik D. Demaine, Matias Korman, Joseph Mitchell, Yota Otachi, Marcel Roeloffzen, Ryuhei Uehara, Yushi Uno, and Andre van Renssen. Symmetric Assembly Puzzles are Hard, Beyond a Few Pieces. In The 18th Japan Conference on Discrete and Computational Geometry and Graphs (JCDCGG 2015), 2015.
Joseph Malkevitch. Problem 707, solution by M. Goldberg. Math. Mag., 42:158, 1969.
Joseph Malkevitch. Tiling Convex Polygons with Equilateral Triangles and Squares. Annals of the New York Academy of Science, 440:299-303, 1985.
Jerry Slocum and Jacob Botermans. The Tangram Book: The Story of the Chinese Puzzle with Over 2000 Puzzles to Solve. Sterling Publishing, 2004.
M. Uematsu. Personal communication. 2015.
Fu Traing Wang and Chuan-Chih Hsiung. A Theorem on the Tangram. The American Mathematical Monthly, 49(9):596-599, 1942.
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How to Solve the Cake-Cutting Problem in Sublinear Time
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.
sublinear-time algorithms
cake-cutting problem
simple fair
preassign
approximation
21:1-21:15
Regular Paper
Hiro
Ito
Hiro Ito
Takahiro
Ueda
Takahiro Ueda
10.4230/LIPIcs.FUN.2016.21
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
Threes!, Fives, 1024!, and 2048 are Hard
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.
algorithmic combinatorial game theory
22:1-22:14
Regular Paper
Stefan
Langerman
Stefan Langerman
Yushi
Uno
Yushi Uno
10.4230/LIPIcs.FUN.2016.22
Ahmed Abdelkader, Aditya Acharya, and Philip Dasler. On the complexity of slide-and-merge games. CoRR, abs/1501.03837, 2015. URL: http://arxiv.org/abs/1501.03837,
http://arxiv.org/abs/1501.03837
Ahmed Abdelkader, Aditya Acharya, and Philip Dasler. 2048 without new tiles is still hard. Proceedings of the 8th International Conference on Fun with Algorithms, LIPICS volume 49, 2016.
Ron Breukelaar, Erik D. Demaine, Susan Hohenberger, Hendrik Jan Hoogeboom, Walter A. Kosters, and David Liben-Nowell. Tetris is hard, even to approximate. International Journal of Computational Geometry and Applications, 14(1-2):41-68, 2004.
Christopher Chen. 2048 is in NP. http://blog.openendings.net/2014/03/2048-is-in- http://blog.openendings.net/2014/03/2048-is-in-np.html, March 2014.
http://blog.openendings.net/2014/03/2048-is-in-np.html
Gabriele Cirulli. 2048. http://gabrielecirulli.github.io/2048/, March 2014.
http://gabrielecirulli.github.io/2048/
Erik D. Demaine and Robert A. Hearn. Playing games with algorithms: Algorithmic combinatorial game theory. In Michael H. Albert and Richard J. Nowakowski, editors, Games of No Chance 3, volume 56 of Mathematical Sciences Research Institute Publications, pages 3-56. Cambridge University Press, 2009. URL: http://arxiv.org/abs/cs.CC/0106019.
http://arxiv.org/abs/cs.CC/0106019
Rodney G. Downey and Michael R. Fellows. Parameterized complexity. Springer Heidelberg, 1999.
Rahul Mehta. 2048 is (PSPACE) hard, but sometimes easy. CoRR, abs/1408.6315, 2014. URL: http://arxiv.org/abs/1408.6315,
http://arxiv.org/abs/1408.6315
Phenomist. 2048 variants. http://phenomist.wordpress.com/2048-variants/, 2014.
http://phenomist.wordpress.com/2048-variants/
QuadmasterXLII. Solve a deterministic version of 2048 using the fewest bytes. http://codegolf.stackexchange.com/questions/24885/solve-a-deterministic- http://codegolf.stackexchange.com/questions/24885/solve-a-deterministic-version-of-2048-using-the-fewest-bytes, 2014.
http://codegolf.stackexchange.com/questions/24885/solve-a-deterministic-version-of-2048-using-the-fewest-bytes
A.J. Richardson. Evil 2048. http://aj-r.github.io/Evil-2048/, March 2014.
http://aj-r.github.io/Evil-2048/
Saming. 2048. http://saming.fr/p/2048/, March 2014.
http://saming.fr/p/2048/
Asher Vollmer, Greg Wohlwend, and Jimmy Hinson. Threes! http://asherv.com/threes/, January 2014.
http://asherv.com/threes/
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An Arithmetic for Rooted Trees
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".
Arithmetic
Rooted tree
Prime tree
Tree equation
23:1-23:14
Regular Paper
Fabrizio
Luccio
Fabrizio Luccio
10.4230/LIPIcs.FUN.2016.23
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Two Dots is NP-complete
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.
combinatorial game theory
NP-complete
perfect information
puzzle
24:1-24:12
Regular Paper
Neeldhara
Misra
Neeldhara Misra
10.4230/LIPIcs.FUN.2016.24
Ahmed Abdelrazek, Aditya Acharya, and Philip Dasler. 2048 without new tiles is still hard. In Proceedings of the 8th International Conference on Fun with Algorithms (To Appear), 2016.
Aaron B. Adcock, Erik D. Demaine, Martin L. Demaine, Michael P. O'Brien, Felix Reidl, Fernando Sanchez Villaamil, and Blair D. Sullivan. Zig-zag numberlink is NP-complete. JIP, 23(3):239-245, 2015.
Matteo Almanza, Stefano Leucci, and Alessandro Panconesi. Trainyard is NP-hard. In Proceedings of the 8th International Conference on Fun with Algorithms (To Appear), 2016.
Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015.
Demaine. Playing games with algorithms: Algorithmic combinatorial game theory. In MFCS: Symposium on Mathematical Foundations of Computer Science, 2001.
Fedor V. Fomin, Pinar Heggernes, and Erik Jan van Leeuwen. Making life easier for firefighters. In Proceedings of the 6th International Conference on Fun with Algorithms, volume 7288, pages 177-188. Springer, 2012.
M. R. Garey and D. S. Johnson. Computers and Intractability. Freeman, San Francisco, 1979.
Stefan Langerman and Yushi Uno. Threes!, Fives, 1024!, and 2048 are hard. In Proceedings of the 8th International Conference on Fun with Algorithms (To Appear), 2016.
Toby Walsh. Candy crush is NP-hard. CoRR, abs/1403.1911, 2014. URL: http://arxiv.org/abs/1403.1911.
http://arxiv.org/abs/1403.1911
Creative Commons Attribution 3.0 Unported license
https://creativecommons.org/licenses/by/3.0/legalcode
This House Proves That Debating is Harder Than Soccer
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.
complexity
elimination games
soccer
debating
25:1-25:14
Regular Paper
Stefan
Neumann
Stefan Neumann
Andreas
Wiese
Andreas Wiese
10.4230/LIPIcs.FUN.2016.25
Mario Basler. Das habe ich ihm dann auch verbal …. Fussballzitate.com. Accessed: 2016-04-22. URL: http://www.fussballzitate.com/zitate/das-habe-ich-ihm-dann-auch-verbal-gesagt.html.
http://www.fussballzitate.com/zitate/das-habe-ich-ihm-dann-auch-verbal-gesagt.html
Thorsten Bernholt, Alexander Gülich, Thomas Hofmeister, and Niels Schmitt. Football elimination is hard to decide under the 3-point-rule. In Proceedings of the 24th International Symposium on Mathematical Foundations of Computer Science, MFCS'99, pages 410-418, London, UK, UK, 1999. Springer-Verlag. URL: http://dx.doi.org/10.1007/3-540-48340-3_37.
http://dx.doi.org/10.1007/3-540-48340-3_37
Celebrating 200 years of free speech and the art of debating. The Cambridge Union Society. Accessed: 2015-12-18. URL: https://cus.org/.
https://cus.org/
Mark Cartwright. Athenian democracy. Ancient History Encyclopedia. Accessed: 2015-12-18. URL: http://www.ancient.eu/Athenian_Democracy.
http://www.ancient.eu/Athenian_Democracy
Katarína Cechlárová, Eva Potpinková, and Ildikó Schlotter. Refining the complexity of the sports elimination problem. Discrete Applied Mathematics, 2015.
Nick Cholst. Why 'three points for a win' is a loss for football - a closer look into one of the most important rule changes in football history. Cafè Futebol. Accessed: 2015-12-18. URL: http://www.cafefutebol.net/2013/09/11/why-three-points-for-a-win-is-a-loss-for-football-a-closer-look-into-one-of-the-most-important-rules-in-football-history/.
http://www.cafefutebol.net/2013/09/11/why-three-points-for-a-win-is-a-loss-for-football-a-closer-look-into-one-of-the-most-important-rules-in-football-history/
Kelly Phillips Erb. Numerous FIFA Officials Arrested in Massive Corruption Scheme Tied to World Cup, Other Tournaments. Forbes. Accessed: 2015-12-18. URL: http://www.forbes.com/sites/kellyphillipserb/2015/05/27/numerous-fifa-officials-arrested-in-massive-corruption-scheme-tied-world-cup-other-tournaments/.
http://www.forbes.com/sites/kellyphillipserb/2015/05/27/numerous-fifa-officials-arrested-in-massive-corruption-scheme-tied-world-cup-other-tournaments/
Handbook. FIDE. Accessed: 2015-12-18. URL: https://www.fide.com/fide/handbook.html?id=18&view=category.
https://www.fide.com/fide/handbook.html?id=18&view=category
2014 FIFA World Cup reached 3.2 billion viewers, one billion watched final. Fédération Internationale de Football Association. Accessed: 2015-12-18. URL: http://www.fifa.com/worldcup/news/y=2015/m=12/news=2014-fifa-world-cuptm-reached-3-2-billion-viewers-one-billion-watched--2745519.html.
http://www.fifa.com/worldcup/news/y=2015/m=12/news=2014-fifa-world-cuptm-reached-3-2-billion-viewers-one-billion-watched--2745519.html
Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman &Co., New York, NY, USA, 1979.
Gusfield and Martel. The structure and complexity of sports elimination numbers. Algorithmica, 32(1):73-86, 2002. URL: http://dx.doi.org/10.1007/s00453-001-0074-y.
http://dx.doi.org/10.1007/s00453-001-0074-y
Marting Luther King Jr. Martin Luther King I Have a Dream Speech. American Rhetoric. Accessed: 2015-12-18. URL: http://www.americanrhetoric.com/speeches/mlkihaveadream.htm.
http://www.americanrhetoric.com/speeches/mlkihaveadream.htm
Walter Kern and Daniël Paulusma. The new FIFA rules are hard: complexity aspects of sports competitions. Discrete Applied Mathematics, 108(3):317-323, 2001. URL: http://dx.doi.org/10.1016/S0166-218X(00)00241-9.
http://dx.doi.org/10.1016/S0166-218X(00)00241-9
Walter Kern and Daniël Paulusma. The computational complexity of the elimination problem in generalized sports competitions. Discrete Optimization, 1(2):205-214, 2004. URL: http://dx.doi.org/10.1016/j.disopt.2003.12.003.
http://dx.doi.org/10.1016/j.disopt.2003.12.003
S. Thomas McCormick. Fast algorithms for parametric scheduling come from extensions to parametric maximum flow. In Proceedings of the Twenty-eighth Annual ACM Symposium on Theory of Computing, STOC'96, pages 319-328, New York, NY, USA, 1996. ACM. URL: http://dx.doi.org/10.1145/237814.237978.
http://dx.doi.org/10.1145/237814.237978
Andreas Möller. "Hauptsache Italien!" Legendäre Fußball-Sprüche. tz München. Accessed: 2016-04-22. URL: http://www.tz.de/sport/fussball/hauptsache-italien-legendaere-fussball-sprueche-fotostrecke-zr-3286394.html.
http://www.tz.de/sport/fussball/hauptsache-italien-legendaere-fussball-sprueche-fotostrecke-zr-3286394.html
Philip Oltermann. Angela Merkel the mascot: German chancellor’s relationship with the team. The Guardian. Accessed: 2015-12-18. URL: http://www.theguardian.com/world/2014/jul/14/angela-merkel-chancellor-german-team-relationship.
http://www.theguardian.com/world/2014/jul/14/angela-merkel-chancellor-german-team-relationship
Dömötör Pálvölgyi. Deciding soccer scores and partial orientations of graphs. Acta Univ. Sapientiae, Math, 1(1):35-42, 2009.
Stuart Pearce. Stuart Pearce quotes. Think Exist. Accessed: 2016-04-22. URL: http://thinkexist.com/quotation/i-can-see-the-carrot-at-the-end-of-the-tunnel/539152.html.
http://thinkexist.com/quotation/i-can-see-the-carrot-at-the-end-of-the-tunnel/539152.html
Ronaldo. Ronaldo quotes. Think Exist. Accessed: 2016-04-22. URL: http://thinkexist.com/quotation/we-lost-because-we-didn-t-win/539102.html.
http://thinkexist.com/quotation/we-lost-because-we-didn-t-win/539102.html
B. L. Schwartz. Possible winners in partially completed tournaments. SIAM Review, 8(3):302-308, 1966. URL: http://dx.doi.org/10.1137/1008062.
http://dx.doi.org/10.1137/1008062
Jürgen Voges. Bundeskanzler: Als Gerhard Schröder noch den Rasen pflügte. Stern. Accessed: 2015-12-18. URL: http://www.stern.de/politik/deutschland/bundeskanzler-als-gerhard-schroeder-noch-den-rasen-pfluegte-3067112.html.
http://www.stern.de/politik/deutschland/bundeskanzler-als-gerhard-schroeder-noch-den-rasen-pfluegte-3067112.html
Kevin D. Wayne. A new property and a faster algorithm for baseball elimination. SIAM Journal on Discrete Mathematics, 14(2):223-229, 2001. URL: http://dx.doi.org/10.1137/S0895480198348847.
http://dx.doi.org/10.1137/S0895480198348847
Rules. World Debating News. Accessed: 2015-12-18. URL: http://worlddebating.blogspot.ie/p/rules.html.
http://worlddebating.blogspot.ie/p/rules.html
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https://creativecommons.org/licenses/by/3.0/legalcode