Jump Markov chains and rejection-free Metropolis algorithms

Solving Generalized Polyomino Puzzles Using the Ising Model

In the polyomino puzzle, the aim is to fill a finite space using several polyomino pieces with no overlaps or blanks. Because it is an NP-complete combinatorial optimization problem, various probabilistic and approximated approaches have been applied to find solutions. Several previous studies embedded the polyomino puzzle in a QUBO problem, where the original objective function and constraints are transformed into the Hamiltonian function of the simulated Ising model. A solution to the puzzle is obtained by searching for a ground state of Hamiltonian by simulating the dynamics of the multiple-spin system. However, previous methods could solve only tiny polyomino puzzles considering a few combinations because their Hamiltonian designs were not efficient. We propose an improved Hamiltonian design that introduces new constraints and guiding terms to weakly encourage favorable spins and pairs in the early stages of computation. The proposed model solves the pentomino puzzle represented by approximately 2000 spins with >90% probability. Additionally, we extended the method to a generalized problem where each polyomino piece could be used zero or more times and solved it with approximately 100% probability. The proposed method also appeared to be effective for the 3D polycube puzzle, which is similar to applications in fragment-based drug discovery.