A Lower Bound for the n-queens Problem

@inproceedings{Simkin2022ALB,
  title={A Lower Bound for the n-queens Problem},
  author={Michael Simkin and Zur Luria},
  booktitle={SODA},
  year={2022}
}
The n-queens puzzle is to place n mutually non-attacking queens on an n × n chess board. We present a simple randomized algorithm to construct such configurations. We first use a random greedy algorithm to construct an approximate toroidal n-queens configuration. In this well-known variant the diagonals wrap around the board from left to right and from top to bottom. We show that with high probability this algorithm succeeds in placing (1 − o(1))n queens on the board. Furthermore, by making… 

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References

SHOWING 1-10 OF 18 REFERENCES
Large girth approximate Steiner triple systems
TLDR
This work shows existence of approximate Steiner triple systems with arbitrary high girth, for any fixed g-element vertex-set, and shows that a natural constrained random process typically produces a partial Steiners triple system with (1/6-o(1))n^2 triples and girth larger than \ell.
The existence of designs via iterative absorption
In a recent breakthrough, Keevash proved the Existence conjecture for combinatorial designs, which has its roots in the 19th century. We give a new proof, based on the method of iterative absorption.
On a Conjecture of Erdős on Locally Sparse Steiner Triple Systems
TLDR
This work proves the Erdős conjecture that one can find so-called ‘sparse’ Steiner triple systems with arbitrary parameters asymptotically by analysing a natural generalization of the triangle removal process.
A survey of known results and research areas for n-queens
Solving Large-Scale Constraint-Satisfaction and Scheduling Problems Using a Heuristic Repair Method
TLDR
A theoretical analysis is presented to explain why the heuristic method for solving large-scale constraint satisfaction and scheduling problems works so well on certain types of problems and to predict when it is likely to be most effective.
The n-queens problem
Random Graph Processes with Degree Restrictions
TLDR
It is proved that if n → ∞ with d fixed, then with probability tending to 1, the final result of this process is a graph with ⌊ nd / 2⌋ edges.
New bounds on the number of n-queens configurations
In how many ways can $n$ queens be placed on an $n \times n$ chessboard so that no two queens attack each other? This is the famous $n$-queens problem. Let $Q(n)$ denote the number of such
Counting designs
We give estimates on the number of combinatorial designs, which prove (and generalise) a conjecture of Wilson from 1974 on the number of Steiner Triple Systems. This paper also serves as an
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