eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
0
0
10.4230/LIPIcs.FUN.2018
article
LIPIcs, Volume 100, FUN'18, Complete Volume
Ito, Hiro
Leonardi, Stefano
Pagli, Linda
Prencipe, Giuseppe
LIPIcs, Volume 100, FUN'18, Complete Volume
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018/LIPIcs.FUN.2018.pdf
Theory of computation, Complexity classes, Algorithm design techniques, Computability, Approximation algorithms analysis, Mathematics of computing
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
0:i
0:xi
10.4230/LIPIcs.FUN.2018.0
article
Front Matter, Table of Contents, Preface, Conference Organization
Ito, Hiro
Leonardi, Stefano
Pagli, Linda
Prencipe, Giuseppe
Front Matter, Table of Contents, Preface, Conference Organization
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.0/LIPIcs.FUN.2018.0.pdf
Front Matter
Table of Contents
Preface
Conference Organization
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
1:1
1:1
10.4230/LIPIcs.FUN.2018.1
article
Mind the Gap (Invited Paper)
Farach-Colton, Martín
1
https://orcid.org/0000-0003-3616-7788
Rutgers University, Department of Computer Science, Piscataway, NJ 08854, USA
As a New Yorker, I'm painfully aware of space. There is, after all, nothing more luxurious than empty space! So when it comes to algorithms, I'm all in favor of leaving holes in my data structures. In this talk, I'll explore the advantages of pampering algorithms with some much needed breathing room.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.1/LIPIcs.FUN.2018.1.pdf
library sort
Italian island
packed memory arrays
weight balanced trees
Italians know how to throw a conference
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
2:1
2:8
10.4230/LIPIcs.FUN.2018.2
article
Evolution of Impossible Objects (Invited Paper)
Sugihara, Kokichi
1
Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan, http://www.isc.meiji.ac.jp/~kokichis/
Impossible objects - 3D objects that can create a visual effect that seems impossible - can be classified by generation based on the order in which they were discovered or produced. The first generation consists of objects whose appearance when observed from a certain viewpoint matches a picture of an impossible object. Many such objects can be created, as there are multiple 3D objects that will project the same two-dimensional picture, including shapes that the human vision system is unable to perceive. The gap between the mathematical and the psychological can also create other types of "impossible" visual effects. Impossible objects are here classified into seven groups.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.2/LIPIcs.FUN.2018.2.pdf
Ambiguous cylinder
anomalous picture
impossible motion
impossible object
optical illusion
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
3:1
3:21
10.4230/LIPIcs.FUN.2018.3
article
Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible
Abel, Zachary
1
Bosboom, Jeffrey
2
Demaine, Erik D.
2
Hamilton, Linus
3
Hesterberg, Adam
3
Kopinsky, Justin
2
Lynch, Jayson
2
Rudoy, Mikhail
2
MIT EECS Department, 50 Vassar St., Cambridge, MA 02139, USA
MIT CSAIL, 32 Vassar Street, Cambridge, MA 02139, USA
MIT Mathematics Department, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
We analyze the computational complexity of the many types of pencil-and-paper-style puzzles featured in the 2016 puzzle video game The Witness. In all puzzles, the goal is to draw a path in a rectangular grid graph from a start vertex to a destination vertex. The different puzzle types place different constraints on the path: preventing some edges from being visited (broken edges); forcing some edges or vertices to be visited (hexagons); forcing some cells to have certain numbers of incident path edges (triangles); or forcing the regions formed by the path to be partially monochromatic (squares), have exactly two special cells (stars), or be singly covered by given shapes (polyominoes) and/or negatively counting shapes (antipolyominoes). We show that any one of these clue types (except the first) is enough to make path finding NP-complete ("witnesses exist but are hard to find"), even for rectangular boards. Furthermore, we show that a final clue type (antibody), which necessarily "cancels" the effect of another clue in the same region, makes path finding Sigma_2-complete ("witnesses do not exist"), even with a single antibody (combined with many anti/polyominoes), and the problem gets no harder with many antibodies.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.3/LIPIcs.FUN.2018.3.pdf
video games
puzzles
hardness
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
4:1
4:13
10.4230/LIPIcs.FUN.2018.4
article
Tracks from hell - when finding a proof may be easier than checking it
Almanza, Matteo
1
Leucci, Stefano
2
https://orcid.org/0000-0002-8848-7006
Panconesi, Alessandro
1
Dipartimento di Informatica, Sapienza Università di Roma, Italy.
Institute of Theoretical Computer Science, ETH Zürich, Switzerland.
We consider the popular smartphone game Trainyard: a puzzle game that requires the player to lay down tracks in order to route colored trains from departure stations to suitable arrival stations. While it is already known [Almanza et al., FUN 2016] that the problem of finding a solution to a given Trainyard instance (i.e., game level) is NP-hard, determining the computational complexity of checking whether a candidate solution (i.e., a track layout) solves the level was left as an open problem. In this paper we prove that this verification problem is PSPACE-complete, thus implying that Trainyard players might not only have a hard time finding solutions to a given level, but they might even be unable to efficiently recognize them.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.4/LIPIcs.FUN.2018.4.pdf
puzzle games
solitaire games
Trainyard
verification
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
5:1
5:13
10.4230/LIPIcs.FUN.2018.5
article
How Bad is the Freedom to Flood-It?
Belmonte, Rémy
1
Khosravian Ghadikolaei, Mehdi
2
Kiyomi, Masashi
3
Lampis, Michael
2
Otachi, Yota
4
https://orcid.org/0000-0002-0087-853X
The University of Electro-Communications, Chofu, Tokyo, Japan
Université Paris-Dauphine, PSL Research University, CNRS, UMR, LAMSADE, 75016 Paris, France
Yokohama City University, Yokohama, Japan
Kumamoto University, Kumamoto, Japan
Fixed-Flood-It and Free-Flood-It are combinatorial problems on graphs that generalize a very popular puzzle called Flood-It. Both problems consist of recoloring moves whose goal is to produce a monochromatic ("flooded") graph as quickly as possible. Their difference is that in Free-Flood-It the player has the additional freedom of choosing the vertex to play in each move. In this paper, we investigate how this freedom affects the complexity of the problem. It turns out that the freedom is bad in some sense. We show that some cases trivially solvable for Fixed-Flood-It become intractable for Free-Flood-It. We also show that some tractable cases for Fixed-Flood-It are still tractable for Free-Flood-It but need considerably more involved arguments. We finally present some combinatorial properties connecting or separating the two problems. In particular, we show that the length of an optimal solution for Fixed-Flood-It is always at most twice that of Free-Flood-It, and this is tight.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.5/LIPIcs.FUN.2018.5.pdf
flood-filling game
parameterized complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
6:1
6:21
10.4230/LIPIcs.FUN.2018.6
article
How long does it take for all users in a social network to choose their communities?
Bermond, Jean-Claude
1
Chaintreau, Augustin
2
Ducoffe, Guillaume
3
Mazauric, Dorian
4
Université Côte d'Azur, CNRS, Inria, I3S, France
Columbia University in the City of New York, USA
National Institute for Research and Development in Informatics and Research Institute of the University of Bucharest, Bucureşti, România
Université Côte d'Azur, Inria, France
We consider a community formation problem in social networks, where the users are either friends or enemies. The users are partitioned into conflict-free groups (i.e., independent sets in the conflict graph G^- =(V,E) that represents the enmities between users). The dynamics goes on as long as there exists any set of at most k users, k being any fixed parameter, that can change their current groups in the partition simultaneously, in such a way that they all strictly increase their utilities (number of friends i.e., the cardinality of their respective groups minus one). Previously, the best-known upper-bounds on the maximum time of convergence were O(|V|alpha(G^-)) for k <= 2 and O(|V|^3) for k=3, with alpha(G^-) being the independence number of G^-. Our first contribution in this paper consists in reinterpreting the initial problem as the study of a dominance ordering over the vectors of integer partitions. With this approach, we obtain for k <= 2 the tight upper-bound O(|V| min {alpha(G^-), sqrt{|V|}}) and, when G^- is the empty graph, the exact value of order ((2|V|)^{3/2})/3. The time of convergence, for any fixed k >= 4, was conjectured to be polynomial [Escoffier et al., 2012][Kleinberg and Ligett, 2013]. In this paper we disprove this. Specifically, we prove that for any k >= 4, the maximum time of convergence is an Omega(|V|^{Theta(log{|V|})}).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.6/LIPIcs.FUN.2018.6.pdf
communities
social networks
integer partitions
coloring games
graphs
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
7:1
7:15
10.4230/LIPIcs.FUN.2018.7
article
On the Complexity of Two Dots for Narrow Boards and Few Colors
Bilò, Davide
1
https://orcid.org/0000-0003-3169-4300
Gualà, Luciano
2
https://orcid.org/0000-0001-6976-5579
Leucci, Stefano
3
https://orcid.org/0000-0002-8848-7006
Misra, Neeldhara
4
https://orcid.org/0000-0003-1727-5388
University of Sassari, Italy
University of Rome "Tor Vergata", Italy
ETH Zürich, Switzerland
Indian Institute of Technology, Gandhinagar
Two Dots is a popular single-player puzzle video game for iOS and Android. A level of this game consists of a grid of colored dots. The player connects two or more adjacent dots, removing them from the grid and causing the remaining dots to fall, as if influenced by gravity. One special move, which is frequently a game-changer, consists of connecting a cycle of dots: this removes all the dots of the given color from the grid. The goal is to remove a certain number of dots of each color using a limited number of moves. The computational complexity of Two Dots has already been addressed in [Misra, FUN 2016], where it has been shown that the general version of the problem is NP-complete. Unfortunately, the known reductions produce Two Dots levels having both a large number of colors and many columns. This does not completely match the spirit of the game, where, on the one hand, only few colors are allowed, and on the other hand, the grid of the game has only a constant number of columns. In this paper, we partially fill this gap by assessing the computational complexity of Two Dots instances having a small number of colors or columns. More precisely, we show that Two Dots is hard even for instances involving only 3 colors or 2 columns. As a contrast, we also prove that the problem can be solved in polynomial-time on single-column instances with a constant number of goals.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.7/LIPIcs.FUN.2018.7.pdf
puzzle
NP-complete
perfect information
combinatorial game theory
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
8:1
8:15
10.4230/LIPIcs.FUN.2018.8
article
On the PSPACE-completeness of Peg Duotaire and other Peg-Jumping Games
Bilò, Davide
1
https://orcid.org/0000-0003-3169-4300
Gualà, Luciano
2
https://orcid.org/0000-0001-6976-5579
Leucci, Stefano
3
https://orcid.org/0000-0002-8848-7006
Proietti, Guido
4
https://orcid.org/0000-0003-1009-5552
Rossi, Mirko
2
Dipartimento di Scienze Umanistiche e Sociali, University of Sassari, Italy.
Dipartimento di Ingegneria dell'Impresa, University of Rome "Tor Vergata", Italy.
Institute of Theoretical Computer Science, ETH Zürich, Switzerland.
Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, University of L'Aquila, Italy, and Istituto di Analisi dei Sistemi ed Informatica, CNR, Roma, Italy.
Peg Duotaire is a two-player version of the classical puzzle called Peg Solitaire. Players take turns making peg-jumping moves, and the first player which is left without available moves loses the game. Peg Duotaire has been studied from a combinatorial point of view and two versions of the game have been considered, namely the single- and the multi-hop variant. On the other hand, understanding the computational complexity of the game is explicitly mentioned as an open problem in the literature. We close this problem and prove that both versions of the game are PSPACE-complete. We also prove the PSPACE-completeness of other peg-jumping games where two players control pegs of different colors.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.8/LIPIcs.FUN.2018.8.pdf
peg duotaire
pspace-completeness
peg solitaire
two-player games
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
9:1
9:10
10.4230/LIPIcs.FUN.2018.9
article
On the Exact Complexity of Polyomino Packing
Bodlaender, Hans L.
1
van der Zanden, Tom C.
2
Department of Computer Science, Utrecht University, Utrecht, The Netherlands and Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands
Department of Computer Science, Utrecht University, Utrecht, The Netherlands
We show that the problem of deciding whether a collection of polyominoes, each fitting in a 2 x O(log n) rectangle, can be packed into a 3 x n box does not admit a 2^{o(n/log{n})}-time algorithm, unless the Exponential Time Hypothesis fails. We also give an algorithm that attains this lower bound, solving any instance of polyomino packing with total area n in 2^{O(n/log{n})} time. This establishes a tight bound on the complexity of Polyomino Packing, even in a very restricted case. In contrast, for a 2 x n box, we show that the problem can be solved in strongly subexponential time.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.9/LIPIcs.FUN.2018.9.pdf
polyomino packing
exact complexity
exponential time hypothesis
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
10:1
10:13
10.4230/LIPIcs.FUN.2018.10
article
Kings, Name Days, Lazy Servants and Magic
Boldi, Paolo
1
https://orcid.org/0000-0002-8297-6255
Vigna, Sebastiano
1
https://orcid.org/0000-0002-3257-651X
Università degli Studi di Milano, Italy
Once upon a time, a king had a very, very long list of names of his subjects. The king was also a bit obsessed with name days: every day he would ask his servants to look the list for all persons having their name day. Reading every day the whole list was taking an enormous amount of time to the king's servants. One day, the chancellor had a magnificent idea: he wrote a book with instructions. The number of pages in the book was equal to the number of names, but following the instructions one could find all people having their name day by looking at only a few pages - in fact, as many pages as the length of the name - and just glimpsing at the list. Everybody was happy, but in time the king's servants got lazy: when the name was very long they would find excuses to avoid looking at so many pages, and some name days were skipped. Desperate, the king made a call through its reign, and a fat sorceress answered. There was a way to look at much, much fewer pages using an additional magic book. But sometimes, very rarely, it would not work (magic does not always work). The king accepted the offer, and name days parties restarted. Only, once every a few thousand years, the magic book fails, and the assistants have to go by the chancellor book. So the parties start a bit later. But they start anyway.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.10/LIPIcs.FUN.2018.10.pdf
Suffix trees
suffix arrays
z-fast tries
prefix search
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
11:1
11:21
10.4230/LIPIcs.FUN.2018.11
article
Computational Complexity of Generalized Push Fight
Bosboom, Jeffrey
1
Demaine, Erik D.
1
Rudoy, Mikhail
1
MIT CSAIL, 32 Vassar Street, Cambridge, MA 02139, USA
We analyze the computational complexity of optimally playing the two-player board game Push Fight, generalized to an arbitrary board and number of pieces. We prove that the game is PSPACE-hard to decide who will win from a given position, even for simple (almost rectangular) hole-free boards. We also analyze the mate-in-1 problem: can the player win in a single turn? One turn in Push Fight consists of up to two "moves" followed by a mandatory "push". With these rules, or generalizing the number of allowed moves to any constant, we show mate-in-1 can be solved in polynomial time. If, however, the number of moves per turn is part of the input, the problem becomes NP-complete. On the other hand, without any limit on the number of moves per turn, the problem becomes polynomially solvable again.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.11/LIPIcs.FUN.2018.11.pdf
board games
hardness
mate-in-one
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
12:1
12:17
10.4230/LIPIcs.FUN.2018.12
article
SUPERSET: A (Super)Natural Variant of the Card Game SET
Botler, Fábio
1
Cristi, Andrés
2
Hoeksma, Ruben
3
Schewior, Kevin
2
Tönnis, Andreas
2
Universidad de Valparaíso, Valparaíso, Chile
Universidad de Chile, Santiago, Chile
Universität Bremen, Bremen, Germany
We consider Superset, a lesser-known yet interesting variant of the famous card game Set. Here, players look for Supersets instead of Sets, that is, the symmetric difference of two Sets that intersect in exactly one card. In this paper, we pose questions that have been previously posed for Set and provide answers to them; we also show relations between Set and Superset.
For the regular Set deck, which can be identified with F^3_4, we give a proof for the fact that the maximum number of cards that can be on the table without having a Superset is 9. This solves an open question posed by McMahon et al. in 2016. For the deck corresponding to F^3_d, we show that this number is Omega(1.442^d) and O(1.733^d). We also compute probabilities of the presence of a superset in a collection of cards drawn uniformly at random. Finally, we consider the computational complexity of deciding whether a multi-value version of Set or Superset is contained in a given set of cards, and show an FPT-reduction from the problem for Set to that for Superset, implying W[1]-hardness of the problem for Superset.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.12/LIPIcs.FUN.2018.12.pdf
SET
SUPERSET
card game
cap set
affine geometry
computational complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
13:1
13:13
10.4230/LIPIcs.FUN.2018.13
article
A Cryptographer's Conspiracy Santa
Bultel, Xavier
1
https://orcid.org/0000-0002-8309-8984
Dreier, Jannik
2
https://orcid.org/0000-0002-1026-3360
Dumas, Jean-Guillaume
3
https://orcid.org/0000-0002-2591-172X
Lafourcade, Pascal
1
https://orcid.org/0000-0002-4459-511X
LIMOS, University Clermont Auvergne, Campus des Cézeaux, Aubière, France
Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Université Grenoble Alpes, Laboratoire Jean Kuntzmann, UMR CNRS 5224, 700 avenue centrale, IMAG - CS 40700, 38058 Grenoble cedex 9, France
In Conspiracy Santa, a variant of Secret Santa, a group of people offer each other Christmas gifts, where each member of the group receives a gift from the other members of the group. To that end, the members of the group form conspiracies, to decide on appropriate gifts, and usually divide the cost of each gift among all participants of that conspiracy. This requires to settle the shared expenses per conspiracy, so Conspiracy Santa can actually be seen as an aggregation of several shared expenses problems.
First, we show that the problem of finding a minimal number of transaction when settling shared expenses is NP-complete. Still, there exists good greedy approximations. Second, we present a greedy distributed secure solution to Conspiracy Santa. This solution allows a group of people to share the expenses for the gifts in such a way that no participant learns the price of his gift, but at the same time notably reduces the number of transactions with respect to a naive aggregation. Furthermore, our solution does not require a trusted third party, and can either be implemented physically (the participants are in the same room and exchange money using envelopes) or, virtually, using a cryptocurrency.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.13/LIPIcs.FUN.2018.13.pdf
Secret Santa
Conspiracy Santa
Secure Multi-Party Computation
Cryptocurrency
Physical Cryptography
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
14:1
14:16
10.4230/LIPIcs.FUN.2018.14
article
Cooperating in Video Games? Impossible! Undecidability of Team Multiplayer Games
Coulombe, Michael J.
1
Lynch, Jayson
1
MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar Street, Cambridge, MA 02139, USA
We show the undecidability of whether a team has a forced win in a number of well known video games including: Team Fortress 2, Super Smash Brothers: Brawl, and Mario Kart.To do so, we give a simplification of the Team Computation Game [Hearn and Demaine, 2009] and use that to give an undecidable abstract game on graphs. This graph game framework better captures the geometry and common constraints in many games and is thus a powerful tool for showing their computational complexity.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.14/LIPIcs.FUN.2018.14.pdf
computational complexity
undecidable
team games
imperfect information
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
15:1
15:19
10.4230/LIPIcs.FUN.2018.15
article
A Muffin-Theorem Generator
Cui, Guangqi
1
Dickerson, John
2
Durvasula, Naveen
1
Gasarch, William
3
Metz, Erik
4
Prinz, Jacob
5
Raman, Naveen
6
Smolyak, Daniel
7
Yoo, Sung Hyun
8
Montgomery Blair High School
Department of Computer Science and UMIACS, Univ of MD at College Park
Department of Computer Science, Univ of MD at College Park
Department of Mathematics, Univ of MD at College Park (ugrad)
Department of Physics, Univ of MD at College Park (ugrad)
Richard Montgomery High School
Department of Computer Science (ugrad). Univ of MD at College Park
Bergen County Academies (a High School)
Consider the following FUN problem. Given m,s you want to divide m muffins among s students so that everyone gets m/(s) muffins; however, you want to maximize the minimum piece so that nobody gets crumbs. Let f(m,s) be the size of the smallest piece in an optimal procedure.
We study the case where ceil(2m/s)=3 because (1) many of our hardest open problems were of this form until we found this method, (2) we have used the technique to generate muffin-theorems, and (3) we conjecture this can be used to solve the general case. We give (1) an algorithm to find an upper bound for f(m,s) when ceil(2m/s)(and some ways to speed up that algorithm if certain conjectures are true), (2) an algorithm that uses the information from (1) to try to find a lower bound on f(m,s) (a procedure) which matches the upper bound, (3) an algorithm that uses the information from (1) to generate muffin-theorems, and (4) an algorithm that we think works well in practice to find f(m,s) for any m,s.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.15/LIPIcs.FUN.2018.15.pdf
Fair Division
Theorem Generation
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
16:1
16:20
10.4230/LIPIcs.FUN.2018.16
article
God Save the Queen
Czyzowicz, Jurek
1
Georgiou, Konstantinos
2
Killick, Ryan
3
Kranakis, Evangelos
3
Krizanc, Danny
4
Narayanan, Lata
5
Opatrny, Jaroslav
5
Shende, Sunil
6
Université du Québec en Outaouais, Gatineau, Québec, Canada
Department of Mathematics, Ryerson University, Toronto, Ontario, Canada
School of Computer Science, Carleton University, Ottawa, Ontario, Canada
Department of Mathematics & Comp. Sci., Wesleyan University, Middletown, CT, USA
Department of Comp. Sci. and Software Eng., Concordia University, Montreal, Québec, Canada
Department of Computer Science, Rutgers University, Camden, NJ, USA
Queen Daniela of Sardinia is asleep at the center of a round room at the top of the tower in her castle. She is accompanied by her faithful servant, Eva. Suddenly, they are awakened by cries of "Fire". The room is pitch black and they are disoriented. There is exactly one exit from the room somewhere along its boundary. They must find it as quickly as possible in order to save the life of the queen. It is known that with two people searching while moving at maximum speed 1 anywhere in the room, the room can be evacuated (i.e., with both people exiting) in 1 + (2 pi)/3 + sqrt{3} ~~ 4.8264 time units and this is optimal [Czyzowicz et al., DISC'14], assuming that the first person to find the exit can directly guide the other person to the exit using her voice. Somewhat surprisingly, in this paper we show that if the goal is to save the queen (possibly leaving Eva behind to die in the fire) there is a slightly better strategy. We prove that this "priority" version of evacuation can be solved in time at most 4.81854. Furthermore, we show that any strategy for saving the queen requires time at least 3 + pi/6 + sqrt{3}/2 ~~ 4.3896 in the worst case. If one or both of the queen's other servants (Biddy and/or Lili) are with her, we show that the time bounds can be improved to 3.8327 for two servants, and 3.3738 for three servants. Finally we show lower bounds for these cases of 3.6307 (two servants) and 3.2017 (three servants). The case of n >= 4 is the subject of an independent study by Queen Daniela's Royal Scientific Team.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.16/LIPIcs.FUN.2018.16.pdf
Algorithm
Evacuation
Exit
Disk
Wireless Communication
Queen
Priority
Robots
Search
Servants
Trajectory
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
17:1
17:14
10.4230/LIPIcs.FUN.2018.17
article
Restricted Power - Computational Complexity Results for Strategic Defense Games
de Haan, Ronald
1
https://orcid.org/0000-0003-2023-0586
Wolf, Petra
2
Institute for Logic, Language and Computation, University of Amsterdam, the Netherlands
Wilhelm-Schickard-Institut, University of Tübingen, Germany
We study the game Greedy Spiders, a two-player strategic defense game, on planar graphs and show PSPACE-completeness for the problem of deciding whether one player has a winning strategy for a given instance of the game. We also generalize our results in metatheorems, which consider a large set of strategic defense games. We achieve more detailed complexity results by restricting the possible strategies of one of the players, which leads us to Sigma^p_2- and Pi^p_2-hardness results.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.17/LIPIcs.FUN.2018.17.pdf
Computational complexity
generalized games
metatheorems
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
18:1
18:21
10.4230/LIPIcs.FUN.2018.18
article
Computational Complexity of Motion Planning of a Robot through Simple Gadgets
Demaine, Erik D.
1
Grosof, Isaac
1
Lynch, Jayson
1
Rudoy, Mikhail
1
MIT CSAIL, 32 Vassar Street, Cambridge, MA 02139, USA
We initiate a general theory for analyzing the complexity of motion planning of a single robot through a graph of "gadgets", each with their own state, set of locations, and allowed traversals between locations that can depend on and change the state. This type of setup is common to many robot motion planning hardness proofs. We characterize the complexity for a natural simple case: each gadget connects up to four locations in a perfect matching (but each direction can be traversable or not in the current state), has one or two states, every gadget traversal is immediately undoable, and that gadget locations are connected by an always-traversable forest, possibly restricted to avoid crossings in the plane. Specifically, we show that any single nontrivial four-location two-state gadget type is enough for motion planning to become PSPACE-complete, while any set of simpler gadgets (effectively two-location or one-state) has a polynomial-time motion planning algorithm. As a sample application, our results show that motion planning games with "spinners" are PSPACE-complete, establishing a new hard aspect of Zelda: Oracle of Seasons.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.18/LIPIcs.FUN.2018.18.pdf
PSPACE
hardness
motion planning
puzzles
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
19:1
19:22
10.4230/LIPIcs.FUN.2018.19
article
The Computational Complexity of Portal and Other 3D Video Games
Demaine, Erik D.
1
Lockhart, Joshua
2
Lynch, Jayson
3
MIT CSAIL, 32 Vassar Street, Cambridge, MA 02139, USA
Department of Computer Science, University College London, London, WC1E 6BT, UK
MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar Street, Cambridge, MA 02139, USA
We classify the computational complexity of the popular video games Portal and Portal 2. We isolate individual mechanics of the game and prove NP-hardness, PSPACE-completeness, or pseudo-polynomiality depending on the specific game mechanics allowed. One of our proofs generalizes to prove NP-hardness of many other video games such as Half-Life 2, Halo, Doom, Elder Scrolls, Fallout, Grand Theft Auto, Left 4 Dead, Mass Effect, Deus Ex, Metal Gear Solid, and Resident Evil. These results build on the established literature on the complexity of video games [Aloupis et al., 2014][Cormode, 2004][Forisek, 2010][Viglietta, 2014].
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.19/LIPIcs.FUN.2018.19.pdf
video games
hardness
motion planning
NP
PSPACE
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
20:1
20:12
10.4230/LIPIcs.FUN.2018.20
article
Faster Evaluation of Subtraction Games
Eppstein, David
1
Computer Science Department, University of California, Irvine
Subtraction games are played with one or more heaps of tokens, with players taking turns removing from a single heap a number of tokens belonging to a specified subtraction set; the last player to move wins. We describe how to compute the set of winning heap sizes in single-heap subtraction games (for an input consisting of the subtraction set and maximum heap size n), in time O~(n), where the O~ elides logarithmic factors. For multi-heap games, the optimal game play is determined by the nim-value of each heap; we describe how to compute the nim-values of all heaps of size up to n in time O~(mn), where m is the maximum nim-value occurring among these heap sizes. These time bounds improve naive dynamic programming algorithms with time O(n|S|), because m <=|S| for all such games. We apply these results to the game of subtract-a-square, whose set of winning positions is a maximal square-difference-free set of a type studied in number theory in connection with the Furstenberg-Sárközy theorem. We provide experimental evidence that, for this game, the set of winning positions has a density comparable to that of the densest known square-difference-free sets, and has a modular structure related to the known constructions for these dense sets. Additionally, this game's nim-values are (experimentally) significantly smaller than the size of its subtraction set, implying that our algorithm achieves a polynomial speedup over dynamic programming.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.20/LIPIcs.FUN.2018.20.pdf
subtraction games
Sprague-Grundy theory
nim-values
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
21:1
21:13
10.4230/LIPIcs.FUN.2018.21
article
Making Change in 2048
Eppstein, David
1
Computer Science Department, University of California, Irvine
The 2048 game involves tiles labeled with powers of two that can be merged to form bigger powers of two; variants of the same puzzle involve similar merges of other tile values. We analyze the maximum score achievable in these games by proving a min-max theorem equating this maximum score (in an abstract generalized variation of 2048 that allows all the moves of the original game) with the minimum value that causes a greedy change-making algorithm to use a given number of coins. A widely-followed strategy in 2048 maintains tiles that represent the move number in binary notation, and a similar strategy in the Fibonacci number variant of the game (987) maintains the Zeckendorf representation of the move number as a sum of the fewest possible Fibonacci numbers; our analysis shows that the ability to follow these strategies is intimately connected with the fact that greedy change-making is optimal for binary and Fibonacci coinage. For variants of 2048 using tile values for which greedy change-making is suboptimal, it is the greedy strategy, not the optimal representation as sums of tile values, that controls the length of the game. In particular, the game will always terminate whenever the sequence of allowable tile values has arbitrarily large gaps between consecutive values.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.21/LIPIcs.FUN.2018.21.pdf
2048
change-making problem
greedy algorithm
integer sequences
halting problem
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
22:1
22:13
10.4230/LIPIcs.FUN.2018.22
article
Pick, Pack, & Survive: Charging Robots in a Modern Warehouse based on Online Connected Dominating Sets
Hamann, Heiko
1
Markarian, Christine
2
Meyer auf der Heide, Friedhelm
3
Wahby, Mostafa
1
Institute of Computer Engineering, University of Lübeck, Germany, https://www.iti.uni-luebeck.de
Heinz Nixdorf Institute, Paderborn University, Germany, https://www.uni-paderborn.de
Heinz Nixdorf Institute, Paderborn University , Germany, https://www.uni-paderborn.de
The modern warehouse is partially automated by robots. Instead of letting human workers walk into shelfs and pick up the required stock, big groups of autonomous mobile robots transport the inventory to the workers. Typically, these robots have an electric drive and need to recharge frequently during the day. When we scale this approach up, it is essential to place recharging stations strategically and as soon as needed so that all robots can survive. In this work, we represent a warehouse topology by a graph and address this challenge with the Online Connected Dominating Set problem (OCDS), an online variant of the classical Connected Dominating Set problem [Guha and Khuller, 1998]. We are given an undirected connected graph G = (V, E) and a sequence of subsets of V arriving over time. The goal is to grow a connected subgraph that dominates all arriving nodes and contains as few nodes as possible. We propose an O(log^2 n)-competitive randomized algorithm for OCDS in general graphs, where n is the number of nodes in the input graph. This is the best one can achieve due to Korman's randomized lower bound of Omega(log n log m) [Korman, 2005] for the related Online Set Cover problem [Alon et al., 2003], where n is the number of elements and m is the number of subsets. We also run extensive simulations to show that our algorithm performs well in a simulated warehouse, where the topology of a warehouse is modeled as a randomly generated geometric graph.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.22/LIPIcs.FUN.2018.22.pdf
connected dominating set
online algorithm
competitive analysis
geometric graph
robot warehouse
recharging stations
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
23:1
23:21
10.4230/LIPIcs.FUN.2018.23
article
Selection Via the Bogo-Method - More on the Analysis of Perversely Awful Randomized Algorithms
Holzer, Markus
1
Maurer, Jan-Tobias
1
Institut für Informatik, Universität Giessen, Arndtstr. 2, 35392 Giessen, Germany
We continue our research on perversely awful randomized algorithms, which started nearly a decade ago. Based on the bogo-method we design a bogo-selection algorithm and variants thereof and analyse them with elementary methods. Moreover, practical experiments are performed.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.23/LIPIcs.FUN.2018.23.pdf
selection
bogo-method
combinatorial sums and series,
inverse binomial coefficients
experimental result
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
24:1
24:11
10.4230/LIPIcs.FUN.2018.24
article
Herugolf and Makaro are NP-complete
Iwamoto, Chuzo
1
Haruishi, Masato
1
Ibusuki, Tatsuaki
2
Hiroshima University, Graduate School of Engineering, Higashi-Hiroshima 739-8527, Japan
Hiroshima University, School of Integrated Arts and Sciences, Higashi-Hiroshima 739-8521, Japan
Herugolf and Makaro are Nikoli's pencil puzzles. We study the computational complexity of Herugolf and Makaro puzzles. It is shown that deciding whether a given instance of each puzzle has a solution is NP-complete.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.24/LIPIcs.FUN.2018.24.pdf
Herugolf
Makaro
pencil puzzle
NP-complete
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
25:1
25:13
10.4230/LIPIcs.FUN.2018.25
article
The Fewest Clues Problem of Picross 3D
Kimura, Kei
1
Kamehashi, Takuya
1
Fujito, Toshihiro
1
Department of Computer Science and Engineering, Toyohashi University of Technology, 1-1 Hibarigaoka, Tempaku, Toyohashi, Aichi, Japan
Picross 3D is a popular single-player puzzle video game for the Nintendo DS. It is a 3D variant of Nonogram, which is a popular pencil-and-paper puzzle. While Nonogram provides a rectangular grid of squares that must be filled in to create a picture, Picross 3D presents a rectangular parallelepiped (i.e., rectangular box) made of unit cubes, some of which must be removed to construct an image in three dimensions. Each row or column has at most one integer on it, and the integer indicates how many cubes in the corresponding 1D slice remain when the image is complete. It is shown by Kusano et al. that Picross 3D is NP-complete. We in this paper show that the fewest clues problem of Picross 3D is Sigma_2^P-complete and that the counting version and the another solution problem of Picross 3D are #P-complete and NP-complete, respectively.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.25/LIPIcs.FUN.2018.25.pdf
Puzzle
computational complexity
fewest clues problem
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
26:1
26:14
10.4230/LIPIcs.FUN.2018.26
article
Uniform Distribution On Pachinko
Kitamura, Naoki
1
Kawabata, Yuya
1
Izumi, Taisuke
1
Nagoya Institute of Technology, Syowa-ku, Gokiso-cho, Nagoya, Aichi, 466-8555, Japan
Pachinko is a japanese mechanical gambling game similar to pinball. Recently, Akitaya et al. proposed several mathematical models of Pachinko. A number of pins are spiked in a field. A ball drops from the top-side end of the playfield, and falls down. In the 50-50 model, if the ball hits a pin, it moves to the left or right of the pin with equal probability. An arrangement of pins generates a distribution of the drop probability over all columns. We consider the problem of generating uniform distributions. Akitaya et al. show that (1/2^{{a}})-uniform distribution is possible for {a} in {0,1,2,3,4} and conjectured that it is possible for any positive integer a. In this paper, we show that the conjecture is true by a constructive way.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.26/LIPIcs.FUN.2018.26.pdf
Pachinko
discrete mathematics
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
27:1
27:19
10.4230/LIPIcs.FUN.2018.27
article
The complexity of speedrunning video games
Lafond, Manuel
1
Department of Mathematics and Statistics, University of Ottawa, Canada
Speedrunning is a popular activity in which the goal is to finish a video game as fast as possible. Players around the world spend hours each day on live stream, perfecting their skills to achieve a world record in well-known games such as Super Mario Bros, Castlevania or Mega Man. But human execution is not the only factor in a successful speed run. Some common techniques such as damage boosting or routing require careful planning to optimize time gains. In this paper, we show that optimizing these mechanics is in fact a profound algorithmic problem, as they lead to novel generalizations of the well-known NP-hard knapsack and feedback arc set problems.
We show that the problem of finding the optimal damage boosting locations in a game admits an FPTAS and is FPT in k + r, the number k of enemy types in the game and r the number of health refill locations. However, if the player is allowed to lose a life to regain health, the problem becomes hard to approximate within a factor 1/2 but admits a (1/2 - epsilon)-approximation with two lives. Damage boosting can also be solved in pseudo-polynomial time. As for routing, we show various hardness results, including W[2]-hardness in the time lost in a game, even on bounded treewidth stage graphs. On the positive side, we exhibit an FPT algorithm for stage graphs of bounded treewidth and bounded in-degree.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.27/LIPIcs.FUN.2018.27.pdf
Approximation algorithms
parameterized complexity
video games
knapsack
feedback arc set
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
28:1
28:16
10.4230/LIPIcs.FUN.2018.28
article
Gender-Aware Facility Location in Multi-Gender World
Polishchuk, Valentin
1
Sedov, Leonid
1
Communications and Transport Systems, ITN, Linköping University, Sweden
This interdisciplinary (GS and CS) paper starts from considering the problem of locating restrooms or locker rooms in a privacy-preserving way, i.e., so that while following the path to one's room, one cannot peek into another room; the rooms are meant for a multitude of genders, one room per gender. We then proceed to showing that gender inequality (non-uniform treatment of genders by genders) makes the room placement hard. Finally, we delve into specifics of gender definition and consider locating facilities for the genders in a "perfect" way, i.e., so that navigating to the facilities involves only quick binary decisions; on the way, we indicate that there is room for interpretation the facilities under consideration (we outline several possibilities, depending on the application).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.28/LIPIcs.FUN.2018.28.pdf
visibility
Strahler number
perfect tree
interval graphs
gender studies
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
29:1
29:10
10.4230/LIPIcs.FUN.2018.29
article
Card-Based Zero-Knowledge Proof for Sudoku
Sasaki, Tatsuya
1
Mizuki, Takaaki
2
Sone, Hideaki
2
Graduate School of Information Sciences, Tohoku University, 6--3--09 Aramaki-Aza-Aoba, Aoba, Sendai 980--8579, Japan
Cyberscience Center, Tohoku University, 6--3 Aramaki-Aza-Aoba, Aoba, Sendai 980--8578, Japan
In 2009, Gradwohl, Naor, Pinkas, and Rothblum proposed physical zero-knowledge proof protocols for Sudoku. That is, for a puzzle instance of Sudoku, their excellent protocols allow a prover to convince a verifier that there is a solution to the Sudoku puzzle and that he/she knows it, without revealing any information about the solution. The possible drawback is that the existing protocols have a soundness error with a non-zero probability or need special cards (such as scratch-off cards). Thus, in this study, we propose new protocols to perform zero-knowledge proof for Sudoku that use a normal deck of playing cards and have no soundness error. Our protocols can be easily implemented by humans with a reasonable number of playing cards.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.29/LIPIcs.FUN.2018.29.pdf
Zero-knowledge proof
Card-based cryptography
Sudoku
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
30:1
30:13
10.4230/LIPIcs.FUN.2018.30
article
The Complexity of Escaping Labyrinths and Enchanted Forests
Schwahn, Florian D.
1
Thielen, Clemens
1
https://orcid.org/0000-0003-0897-3571
Department of Mathematics, University of Kaiserslautern, Paul-Ehrlich-Str. 14, D-67663 Kaiserslautern, Germany
The board games The aMAZEing Labyrinth (or simply Labyrinth for short) and Enchanted Forest published by Ravensburger are seemingly simple family games.
In Labyrinth, the players move though a labyrinth in order to collect specific items. To do so, they shift the tiles making up the labyrinth in order to open up new paths (and, at the same time, close paths for their opponents). We show that, even without any opponents, determining a shortest path (i.e., a path using the minimum possible number of turns) to the next desired item in the labyrinth is strongly NP-hard. Moreover, we show that, when competing with another player, deciding whether there exists a strategy that guarantees to reach one's next item faster than one's opponent is PSPACE-hard.
In Enchanted Forest, items are hidden under specific trees and the objective of the players is to report their locations to the king in his castle. Movements are performed by rolling two dice, resulting in two numbers of fields one has to move, where each of the two movements must be executed consecutively in one direction (but the player can choose the order in which the two movements are performed). Here, we provide an efficient polynomial-time algorithm for computing a shortest path between two fields on the board for a given sequence of die rolls, which also has implications for the complexity of problems the players face in the game when future die rolls are unknown.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.30/LIPIcs.FUN.2018.30.pdf
board games
combinatorial game theory
computational complexity
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
31:1
31:13
10.4230/LIPIcs.FUN.2018.31
article
Card-based Protocols Using Triangle Cards
Shinagawa, Kazumasa
1
Mizuki, Takaaki
2
Tokyo Institute of Technology, Tokyo, Japan, Institute of Advanced Industrial Science and Technology (AIST), Tokyo, Japan
Tohoku University, Sendai, Japan
Suppose that three boys and three girls attend a party. Each boy and girl have a crush on exactly one of the three girls and three boys, respectively. The following dilemma arises: On one hand, each person thinks that if there is a mutual affection between a girl and boy, the couple should go on a date the next day. On the other hand, everyone wants to avoid the possible embarrassing situation in which their heart is broken "publicly." In this paper, we solve the dilemma using novel cards called triangle cards. The number of cards required is only six, which is minimal in the case where each player commits their input at the beginning of the protocol. We also construct multiplication and addition protocols based on triangle cards. Combining these protocols, we can securely compute any function f: {0,1,2}^n --> {0,1,2}.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.31/LIPIcs.FUN.2018.31.pdf
Cryptography without computer
Secure computation
Card-based protocols
Triangle cards
Three-valued computation
Secure matching problem
eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2018-06-04
100
32:1
32:15
10.4230/LIPIcs.FUN.2018.32
article
The Power of One Secret Agent
Tamir, Tami
1
School of Computer Science, The Interdisciplinary Center (IDC), Herzliya, Israel
I am a job. In job-scheduling applications, my friends and I are assigned to machines that can process us. In the last decade, thanks to our strong employee committee, and the rise of algorithmic game theory, we are getting more and more freedom regarding our assignment. Each of us acts to minimize his own cost, rather than to optimize a global objective.
My goal is different. I am a secret agent operated by the system. I do my best to lead my fellow jobs to an outcome with a high social cost. My naive friends keep doing the best they can, each of them performs his best-response move whenever he gets the opportunity to do so. Luckily, I am a charismatic guy. I can determine the order according to which the naive jobs perform their best-response moves. In this paper, I analyze my power, formalized as the Price of a Traitor (PoT), in cost-sharing scheduling games - in which we need to cover the cost of the machines that process us.
Starting from an initial Nash Equilibrium (NE) profile, I join the instance and hurt its stability. A sequence of best-response moves is performed until I vanish, leaving the naive jobs in a new NE. For an initial NE assignment, S_0, the PoT measures the ratio between the social cost of a worst NE I can lead the jobs to, starting from S_0, and the social cost of S_0. The PoT of a game is the maximal such ratio among all game instances and initial NE assignments.
My analysis distinguishes between instances with unit- and arbitrary-cost machines, and instances with unit- and arbitrary-length jobs. I give exact bounds on the PoT for each setting, in general and in symmetric games. While it turns out that in most settings my power is really impressive, my task is computationally hard (and also hard to approximate).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol100-fun2018/LIPIcs.FUN.2018.32/LIPIcs.FUN.2018.32.pdf
Job scheduling games
Cost sharing
Equilibrium inefficiency