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A QUBO Formulation for the Generalized LinkedIn Queens and Takuzu/Tango Game

Alejandro Mata Ali, Edgar Mencia

TL;DR

The paper develops a unified QUBO approach to solve generalized N-Queens variants (including LinkedIn Queens) and Takuzu/Tango problems on boards of various shapes, sizes, and constraints. It defines modular QUBO terms for rows, columns, diagonals, and regions, and extends them to handle initial condition fixes, toroidal geometries, and long-distance constraints. Two new problems, Coloured Chess Piece and Max Chess Pieces, are introduced with corresponding QUBO formulations, alongside Tents & Trees and generalized TTP formulations. The work demonstrates that complex constraint systems can be expressed as a finite set of quadratic penalties, enabling execution on quantum annealers or QAOA and providing a foundation for hardware benchmarking and future algorithmic developments.

Abstract

In this paper, we present a QUBO formulation designed to solve a series of generalisations of the LinkedIn queens game, a version of the N-queens problem, for the Takuzu game (or Binairo), for the most recent LinkedIn game, Tango, and for its generalizations. We adapt this formulation for several particular cases of the problem, as Tents & Trees, by trying to optimise the number of variables and interactions, improving the possibility of applying it on quantum hardware by means of Quantum Annealing or the Quantum Approximated Optimization Algorithm (QAOA). We also present two new types of problems, the Coloured Chess Piece Problem and the Max Chess Pieces Problem, with their corresponding QUBO formulations.

A QUBO Formulation for the Generalized LinkedIn Queens and Takuzu/Tango Game

TL;DR

The paper develops a unified QUBO approach to solve generalized N-Queens variants (including LinkedIn Queens) and Takuzu/Tango problems on boards of various shapes, sizes, and constraints. It defines modular QUBO terms for rows, columns, diagonals, and regions, and extends them to handle initial condition fixes, toroidal geometries, and long-distance constraints. Two new problems, Coloured Chess Piece and Max Chess Pieces, are introduced with corresponding QUBO formulations, alongside Tents & Trees and generalized TTP formulations. The work demonstrates that complex constraint systems can be expressed as a finite set of quadratic penalties, enabling execution on quantum annealers or QAOA and providing a foundation for hardware benchmarking and future algorithmic developments.

Abstract

In this paper, we present a QUBO formulation designed to solve a series of generalisations of the LinkedIn queens game, a version of the N-queens problem, for the Takuzu game (or Binairo), for the most recent LinkedIn game, Tango, and for its generalizations. We adapt this formulation for several particular cases of the problem, as Tents & Trees, by trying to optimise the number of variables and interactions, improving the possibility of applying it on quantum hardware by means of Quantum Annealing or the Quantum Approximated Optimization Algorithm (QAOA). We also present two new types of problems, the Coloured Chess Piece Problem and the Max Chess Pieces Problem, with their corresponding QUBO formulations.
Paper Structure (25 sections, 42 equations, 9 figures)

This paper contains 25 sections, 42 equations, 9 figures.

Table of Contents

  1. Introduction
  2. LinkedIn's Queens and N-Queens problem
  3. Problem definition
  4. QUBO resolution
  5. General cases
  6. Number of queens per row/column
  7. Distant diagonals constraint
  8. Number of queens per region
  9. Two possible numbers of queens per region
  10. Toroidal board
  11. Definition of generalised N-queens
  12. N-queens with regions
  13. N-queens with soft regions
  14. Tents & Trees
  15. Chess Piece Problems
  16. ...and 10 more sections

Figures (9)

  • Figure 1: Constraints on a $5\times 5$ problem, such that no queens can be placed in any of the cells marked in red. a) Column constraint, b) Row constraint, c) Diagonal constraint on N-queens, d) Diagonal constraint on LQueens, e) Region constraint.
  • Figure 2: a) Solution of an 8-queen problem, b) Solution of an 8-queen problem from LinkedIn
  • Figure 3: a) Irregular shaped board. b) Regular board with initial condition and its equivalent irregular board without initial condition. The cells in grey are squares that are not on the board.
  • Figure 4: Toroidal board with a queen, where we mark in red what would be the squares that this queen could attack from only one of its 4 diagonals.
  • Figure 5: a) Solution of a rectangular Coloured Chess Piece Problem, b) Solution of a rectangular Max Chess Pieces Problem
  • ...and 4 more figures