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A Muffin-Theorem Generator

Authors Guangqi Cui, John Dickerson, Naveen Durvasula, William Gasarch, Erik Metz, Jacob Prinz, Naveen Raman, Daniel Smolyak, Sung Hyun Yoo



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Author Details

Guangqi Cui
  • Montgomery Blair High School
John Dickerson
  • Department of Computer Science and UMIACS, Univ of MD at College Park
Naveen Durvasula
  • Montgomery Blair High School
William Gasarch
  • Department of Computer Science, Univ of MD at College Park
Erik Metz
  • Department of Mathematics, Univ of MD at College Park (ugrad)
Jacob Prinz
  • Department of Physics, Univ of MD at College Park (ugrad)
Naveen Raman
  • Richard Montgomery High School
Daniel Smolyak
  • Department of Computer Science (ugrad). Univ of MD at College Park
Sung Hyun Yoo
  • Bergen County Academies (a High School)

Cite AsGet BibTex

Guangqi Cui, John Dickerson, Naveen Durvasula, William Gasarch, Erik Metz, Jacob Prinz, Naveen Raman, Daniel Smolyak, and Sung Hyun Yoo. A Muffin-Theorem Generator. In 9th International Conference on Fun with Algorithms (FUN 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 100, pp. 15:1-15:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.FUN.2018.15

Abstract

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
Keywords
  • Fair Division
  • Theorem Generation

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References

  1. Guangiqi Cui, John Dickerson, Naveen Durvasula, William Gasarch, Erik Metz, Jacob Prinz, Naveen Raman, Daniel Smolyak, and Sung Hyun Yoo. Code for muffin problems, 2017. URL: https://github.com/jeprinz/MuffinProblem.
  2. Guangiqi Cui, John Dickerson, Naveen Durvasula, William Gasarch, Erik Metz, Jacob Prinz, Naveen Raman, Daniel Smolyak, and Sung Hyun Yoo. The muffin problem, 2017. URL: https://arxiv.org/abs/1709.02452.
  3. Alan Frank. The muffin problem, 2013. Described to Jeremy Copeland and in the New York Times Numberplay Online Blog URL: wordplay.blogs.nytimes.com/2013/08/19/cake.
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