The differential equation method in Banach spaces and the n-queens problem

Combinatorics Seminar
Friday, January 29, 2021 - 3:00pm for 1 hour (actually 50 minutes)
Location (To receive the password, please email Lutz Warnke
Michael Simkin
Lutz Warnke

The differential equation method is a powerful tool used to study the evolution of random combinatorial processes. By showing that the process is likely to follow the trajectory of an ODE, one can study the deterministic ODE rather than the random process directly. We extend this method to ODEs in infinite-dimensional Banach spaces.
We apply this tool to the classical n-queens problem: Let Q(n) be the number of placements of n non-attacking chess queens on an n x n board. Consider the following random process: Begin with an empty board. For as long as possible choose, uniformly at random, a space with no queens in its row, column, or either diagonal, and place on it a queen. We associate the process with an abstract ODE. By analyzing the ODE we conclude that the process almost succeeds in placing n queens on the board. Furthermore, we can obtain a complete n-queens placement by making only a few changes to the board. By counting the number of choices available at each step we conclude that Q(n) \geq (n/C)^n, for a constant C>0 associated with the ODE. This is optimal up to the value of C.

Based on joint work with Zur Luria.