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The n-queens completion problem.

Stefan Glock1, David Munhá Correia2, Benny Sudakov2

  • 1Institute for Theoretical Studies, ETH, 8092 Zurich, Switzerland.

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Summary
This summary is machine-generated.

A partial configuration of at most n/60 non-attacking queens on an n x n chessboard can always be completed to a full n-queens solution. This research explores the minimum size for guaranteed n-queens completion.

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Area of Science:

  • Combinatorial mathematics
  • Computer science theory
  • Graph theory

Background:

  • The n-queens problem involves placing n non-attacking queens on an n x n chessboard.
  • The n-queens completion problem determines if a partial placement can be extended to a full solution.
  • Extremal questions investigate the minimum size of partial configurations for guaranteed completion.

Purpose of the Study:

  • To determine the minimum size of a partial n-queens configuration that guarantees a completion.
  • To establish bounds for the n-queens completion problem.
  • To explore connections between the n-queens problem and graph theory.

Main Methods:

  • Utilizing rainbow matchings in bipartite graphs.
  • Employing probabilistic arguments.
  • Applying linear programming duality.

Main Results:

  • Demonstrated that any placement of at most n/60 mutually non-attacking queens can be completed.
  • Provided examples of partial configurations with approximately n/4 queens that cannot be completed.
  • Formulated new research questions related to the n-queens completion problem.

Conclusions:

  • Established a tight bound for the guaranteed completion of n-queens configurations.
  • Highlighted the utility of graph-theoretic and probabilistic methods in solving combinatorial problems.
  • Opened avenues for further research into extremal aspects of the n-queens problem.