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Solving Generalized Polyomino Puzzles Using the Ising Model.

Kazuki Takabatake1, Keisuke Yanagisawa1, Yutaka Akiyama1

  • 1Department of Computer Science, School of Computing, Tokyo Institute of Technology, Meguro-ku 152-8550, Tokyo, Japan.

Entropy (Basel, Switzerland)
|March 25, 2022
PubMed
Summary
This summary is machine-generated.

This study introduces an improved Hamiltonian design for solving polyomino puzzles, a complex combinatorial optimization problem. The new method efficiently solves larger puzzles, including 3D polycube puzzles, with high probability.

Keywords:
Hopfield neural networkIsing modelcombinatorial optimizationpolyomino puzzle

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Area of Science:

  • Computational Physics
  • Combinatorial Optimization
  • Artificial Intelligence

Background:

  • Polyomino puzzles are NP-complete combinatorial optimization problems.
  • Existing solutions often use Quantum Annealing (QUBO) and Ising models but struggle with larger puzzle sizes due to inefficient Hamiltonian designs.
  • Previous methods were limited to small-scale polyomino puzzles.

Purpose of the Study:

  • To develop a more efficient Hamiltonian design for solving polyomino puzzles.
  • To improve the probability of finding solutions for larger and more complex polyomino and polycube puzzles.
  • To adapt the method for generalized polyomino problems and explore applications in fields like drug discovery.

Main Methods:

  • The study proposes an improved Hamiltonian design by incorporating new constraints and guiding terms.
  • This design aims to encourage favorable spin configurations in the early stages of computation.
  • The method involves simulating the dynamics of a multiple-spin system to find the ground state of the Hamiltonian.

Main Results:

  • The proposed model successfully solves the pentomino puzzle (approx. 2000 spins) with over 90% probability.
  • A generalized version of the problem, allowing pieces to be used multiple times, was solved with approximately 100% probability.
  • The method demonstrated effectiveness for 3D polycube puzzles, suggesting potential in fragment-based drug discovery.

Conclusions:

  • The enhanced Hamiltonian design significantly improves the solvability of polyomino puzzles.
  • The method is scalable and effective for both 2D and 3D puzzle variations, including generalized versions.
  • This approach shows promise for applications beyond puzzles, such as in computational chemistry and drug discovery.