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March 27, 2026Open Access

Polynomial-Time Solution of Unique Latin Square Completion via Prime Debt Graphs and Exhaustive Cascade

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Authors

JTJoão Teixeira

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Overview

This research demonstrates a polynomial-time method to complete unique Latin squares, highlighting its efficiency and theoretical significance.

Key Points

  • The aim is to establish a polynomial-time solution for unique Latin square completions with a focus on a unique solution case.
  • Utilized house polynomials to represent row and column constraints.
  • Applied a prime encoding for candidate sets based on divisibility.
  • Analyzed alternating cycles in the bipartite matching graph in relation to surplus candidates.
  • Developed a prime debt graph to track candidate displacements.
  • Implemented a three-layer operator for violation detection and candidate elimination.
  • Achieved a solution in O(N^11) time for unique Latin square completions.
  • Demonstrated that local feasible reassignments cannot lead to a global solution under the given constraints.
  • Guaranteed elimination of incorrect candidates with each application of the operator.

Cite This Study

João Teixeira (2026) studied this question.

synapsesocial.com/papers/69c6206115a0a509bde18cd4https://doi.org/10.5281/zenodo.19227581
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