PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Tytuł artykułu

On the hat problem on a graph

Autorzy
Treść / Zawartość
Identyfikatory
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The topic of this paper is the hat problem in which each of n players is uniformly and independently fitted with a blue or red hat. Then everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The aim is to maximize the probability of winning. In this version every player can see everybody excluding himself. We consider such a problem on a graph, where vertices correspond to players, and a player can see each player to whom he is connected by an edge. The solution of the hat problem on a graph is known for trees and for cycles on four or at least nine vertices. In this paper first we give an upper bound on the maximum chance of success for graphs with neighborhood-dominated vertices. Next we solve the problem on unicyclic graphs containing a cycle on at least nine vertices. We prove that the maximum chance of success is one by two. Then we consider the hat problem on a graph with a universal vertex. We prove that there always exists an optimal strategy such that in every case some vertex guesses its color. Moreover, we prove that there exists a graph with a universal vertex for which there exists an optimal strategy such that in some case no vertex guesses its color. We also give some Nordhaus-Gaddum type inequalities.
Rocznik
Strony
285--296
Opis fizyczny
Bibliogr. 26 poz., rys., tab.
Twórcy
  • Gdańsk University of Technology Faculty of Electronics, Telecommunications and Informatics ul. Narutowicza 11/12, 80-233 Gdańsk, Poland, marcin.krzywkowski@gmail.com
Bibliografia
  • [1] G. Aggarwal, A. Fiat, A. Goldberg, J. Hartline, N. Immorlica, M. Sudan, Derandomization of auctions, Proc. of the 37th Annual ACM Symposium on Theory of Computing, ACM, New York, 2005, 619–625.
  • [2] J. Aspnes, R. Beigel, M. Furst, S. Rudich, The expressive power of voting polynomials, Combinatorica 14 (1994), 135–148.
  • [3] R. Beigel, L. Fortnow, F. Stephan, Infinitely-often autoreducible sets, SIAM Journal on Computing 36 (2006), 595–608.
  • [4] Berkeley Riddles, www.ocf.berkeley.edu/wwu/riddles/hard.shtml.
  • [5] M. Bernstein, The hat problem and Hamming codes, MAA Focus, November, 2001, 4–6.
  • [6] W. Blum, Denksport für Hutträger, Die Zeit, May 3, 2001.
  • [7] M. Breit, D. Deshommes, A. Falden, Hats required: perfect and imperfect strategies for the hat problem, The Journal of the Summer Undergraduate Mathematical Science Research Institute, Miami University, Summer 2002.
  • [8] E. Brown, K. Mellinger, Kirkman’s schoolgirls wearing hats and walking through fields of numbers, Mathematics Magazine 82 (2009), 3–15.
  • [9] E. Burke, S. Gustafson, G. Kendall, A puzzle to challenge genetic programming, Genetic Programming, 136–147, Lecture Notes in Computer Science, Springer, 2002.
  • [10] S. Butler, M. Hajianghayi, R. Kleinberg, T. Leighton, Hat guessing games, SIAM Journal on Discrete Mathematics 22 (2008), 592–605.
  • [11] G. Cohen, I. Honkala, S. Litsyn, A. Lobstein, Covering Codes, North Holland, 1997.
  • [12] T. Ebert, Applications of recursive operators to randomness and complexity, Ph.D. Thesis,University of California at Santa Barbara, 1998.
  • [13] T. Ebert, W. Merkle, Autoreducibility of random sets: a sharp bound on the density of guessed bits, Mathematical Foundations of Computer Science 2002, 221–233, Lecture Notes in Computer Science, 2420, Springer, Berlin, 2002.
  • [14] T. Ebert, W. Merkle, H. Vollmer, On the autoreducibility of random sequences, SIAM Journal on Computing 32 (2003), 1542–1569.
  • [15] T. Ebert, H. Vollmer, On the autoreducibility of random sequences, Mathematical Foundations of Computer Science 2000 (Bratislava), 333–342, Lecture Notes in Computer Science, 1893, Springer, Berlin, 2000.
  • [16] U. Feige, You can leave your hat on (if you guess its color), Technical Report MCS04-03, Computer Science and Applied Mathematics, The Weizmann Institute of Science, 2004, 10 pp.
  • [17] W. Guo, S. Kasala, M. Rao, B. Tucker, The hat problem and some variations, Advances in Distribution Theory, Order Statistics, and Inference, Statistics for Industry and Technology, Birkhäuser, Boston, 2007, 459–479.
  • [18] N. Immorlica, Computing with strategic agents, Ph.D. Thesis, Massachusetts Institute of Technology, 2005.
  • [19] M. Krzywkowski, A modified hat problem, Commentationes Mathematicae 50 (2010), 121–126.
  • [20] M. Krzywkowski, Hat problem on a graph, Mathematica Pannonica 21 (2010), 3–21.
  • [21] M. Krzywkowski, Hat problem on the cycle C4, International Mathematical Forum 5 (2010), 205–212.
  • [22] M. Krzywkowski, On the hat problem, its variations, and their applications, Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica 9 (2010), 55–67.
  • [23] M. Krzywkowski, The hat problem on cycles on at least nine vertices, Ars Combinatoria 101 (2011), 3–13.
  • [24] H. Lenstra, G. Seroussi, On hats and other covers, IEEE International Symposium on Information Theory, Lausanne, 2002.
  • [25] J. Poulos, Could you solve this $1 million hat trick?, abcNews, November 29, 2001.
  • [26] S. Robinson, Why mathematicians now care about their hat color, The New York Times, Science Times Section, page D5, April 10, 2001.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-AGHT-0007-0020
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.