A Muffin-Theorem Generator
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.
Fair Division
Theorem Generation
Mathematics of computing~Combinatorial optimization
15:1-15:19
Regular Paper
https://arxiv.org/abs/1709.02452
Guangqi
Cui
Guangqi Cui
Montgomery Blair High School
John
Dickerson
John Dickerson
Department of Computer Science and UMIACS, Univ of MD at College Park
Naveen
Durvasula
Naveen Durvasula
Montgomery Blair High School
William
Gasarch
William Gasarch
Department of Computer Science, Univ of MD at College Park
Erik
Metz
Erik Metz
Department of Mathematics, Univ of MD at College Park (ugrad)
Jacob
Prinz
Jacob Prinz
Department of Physics, Univ of MD at College Park (ugrad)
Naveen
Raman
Naveen Raman
Richard Montgomery High School
Daniel
Smolyak
Daniel Smolyak
Department of Computer Science (ugrad). Univ of MD at College Park
Sung Hyun
Yoo
Sung Hyun Yoo
Bergen County Academies (a High School)
10.4230/LIPIcs.FUN.2018.15
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.
https://github.com/jeprinz/MuffinProblem
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.
https://arxiv.org/abs/1709.02452
Alan Frank. The muffin problem, 2013. Described to Jeremy Copeland and in the New York Times Numberplay Online Blog URL: https://wordplay.blogs.nytimes.com/2013/08/19/cake.
https://wordplay.blogs.nytimes.com/2013/08/19/cake
Guangqi Cui, John Dickerson, Naveen Durvasula, William Gasarch, Erik Metz, Jacob Prinz, Naveen Raman, Daniel Smolyak, and Sung Hyun Yoo
Creative Commons Attribution 3.0 Unported license
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