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arXiv:2407.02091 · Scientific Reports

Efficient bit labeling in factorization machines with annealing for traveling salesman problem

Shota Koshikawa, Aruto Hosaka, Tsuyoshi Yoshida

Published Jul 24, 2025• 0 citations
arXivHTMLPDF

Abstract

To efficiently determine an optimum parameter combination in a large-scale problem, it is essential to convert the parameters into available variables in actual machines. Specifically, quadratic unconstrained binary optimization problems are solved using machine learning, for example, factorization machines with annealing, which convert a raw parameter to binary variables. This study investigates the dependence of the convergence speed and accuracy on the binary labeling method, which can influence the cost function shape and thus the probability of being captured at a local minimum solution. By exemplifying the traveling salesman problem (TSP), we propose and evaluate Gray labeling, which correlates the Hamming distance in binary labels with the traveling distance. Through numerical simulation of the TSP at a limited number of iterations, the Gray labeling shows fewer local minima percentages and shorter traveling distances compared with natural labeling.

DOI
10.1038/s41598-025-10064-4

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To efficiently determine an optimum parameter combination in a large-scale problem, it is essential to convert the parameters into available variables in actual machines. Specifically, quadratic unconstrained binary optimization problems are solved using machine learning, for example, factorization machines with annealing, which convert a raw parameter to binary variables. This study investigates the dependence of the convergence speed and accuracy on the binary labeling method, which can influence the cost function shape and thus the probability of being captured at a local minimum solution. By exemplifying the traveling salesman problem (TSP), we propose and evaluate Gray labeling, which correlates the Hamming distance in binary labels with the traveling distance. Through numerical simulation of the TSP at a limited number of iterations, the Gray labeling shows fewer local minima percentages and shorter traveling distances compared with natural labeling.
In black-box combinatorial optimization, objective evaluations are often expensive, so high quality solutions must be found under a limited budget. Factorization machine with quantum annealing (FMQA) builds a quadratic surrogate model from evaluated samples and optimizes it on an Ising machine. However, FMQA requires binary decision variables, and for nonbinary structures such as integer permutations, the choice of binary encoding strongly affects search efficiency. If the encoding fails to reflect the original neighborhood structure, small Hamming moves may not correspond to meaningful modifications in the original solution space, and constrained problems can yield many infeasible candidates that waste evaluations. Recent work combines FMQA with a binary autoencoder (bAE) that learns a compact binary latent code from feasible solutions, yet the mechanism behind its performance gains is unclear. Using a small traveling salesman problem as an interpretable testbed, we show that the bAE reconstructs feasible tours accurately and, compared with manually designed encodings at similar compression, better aligns tour distances with latent Hamming distances, yields smoother neighborhoods under small bit flips, and produces fewer local optima. These geometric properties explain why bAE+FMQA improves the approximation ratio faster while maintaining feasibility throughout optimization, and they provide guidance for designing latent representations for black-box optimization.
Annealing-based Ising machines have shown promising results in solving combinatorial optimization problems. As a typical class of these problems, however, traveling salesman problems (TSPs) are very challenging to solve due to the constraints imposed on the solution. This article proposes a parallel annealing algorithm for a fully connected Ising machine that significantly improves the accuracy and performance in solving constrained combinatorial optimization problems such as the TSP. Unlike previous parallel annealing algorithms, this improved parallel annealing (IPA) algorithm efficiently solves TSPs using an exponential temperature function with a dynamic offset. Compared with digital annealing (DA) and momentum annealing (MA), the IPA reduces the run time by 44.4 times and 19.9 times for a 14-city TSP, respectively. Large scale TSPs can be more efficiently solved by taking a k-medoids clustering approach that decreases the average travel distance of a 22-city TSP by 51.8% compared with DA and by 42.0% compared with MA. This approach groups neighboring cities into clusters to form a reduced TSP, which is then solved in a hierarchical manner by using the IPA algorithm.
Abstract The traveling salesman problem (TSP) forms the foundation for many logistics and optimization tasks. At the same time, this problem is NP-hard and requires significant computational resources. Quantum computing promises to substantially accelerate computations or increase solution accuracy. The TSP can be conveniently mapped onto the quadratic unconstrained binary optimization (QUBO) formulation, which in turn can be solved using variational quantum eigensolver (VQE) and quantum approximate optimization algorithm (QAOA) methods, but this mapping requires a quadratic number of qubits. The higher-order binary optimization (HOBO) formulation requires significantly fewer qubits, but it contains higher-order terms. In our work, we thoroughly investigate both formulations and provide a practical recipe for solving the TSP. We give a comprehensive study and compare in detail QUBO and HOBO formulations.
When solving a combinatorial optimization problem with Ising machines, we have to formulate it onto an energy function of Ising model. Here, how to determine the penalty coefficients in the energy function is a great concern if it includes constraint terms. In this paper, we focus on a traveling salesman problem (TSP, in short), one of the combinatorial optimization problems with equality constraints. Firstly, we investigate the relationship between the penalty coefficient and the accuracy of solutions in a TSP. Based on it, we propose a method to obtain a TSP quasi-optimum solution, which is called multiple coefficients trial method. In our proposed method, we use an Ising machine to solve a TSP by changing a penalty coefficient every trial, the TSP solutions can converge very fast in total. Compared to naive methods using simulated-annealing-based Ising machines, we confirmed that our proposed method can reduce the total number of annealing iterations to 1/10 to 1/1000 to obtain a quasi-optimum solution in 32-city TSPs.
Ising machines have attracted much attention in recent years because of their potential in solving combinatorial optimization problems that are difficult to be solved using conventional computers. Ising machines solve the optimization problem that can be transformed into a quadratic unconstrained binary optimization (QUBO) problem. In the case of a problem with constraints, named a constraint problem, an objective function and constraint function with penalty coefficients are combined into a single Hamiltonian of a QUBO problem. When solving a constraint problem, the accuracy of the solution decreases compared to a QUBO problem with an objective function only. This is because penalty coefficients are set to excessively large values to prevent constraint violations. As a result, it becomes difficult to minimize the objective function. To solve this problem, this paper proposes a method that partitions constraint functions and reduces the value of the penalty coefficients. The performance evaluation using the traveling salesperson problem (TSP) with one-hot constraints demonstrates that the proposed method can improve the solution accuracy compared to the conventional method.
Many combinatorial optimization problems can be solved by numerically simulating classical nonlinear Hamiltonian systems based on the Ising model. Solving the traveling salesman problem (TSP) using the Ising model requires a quadratically increasing number of spins with strict constraints. Unlike classical simulated annealing, simulated bifurcation (SB) can update the states of spins in parallel. This feature can potentially accelerate the convergence of Hamiltonian in the Ising model by taking advantage of modern multi-core processors. As an improved SB algorithm, the ballistic SB (bSB) algorithm is considered for solving the TSP in this paper. The TSP is converted to an Ising problem with external magnetic fields. bSB is then expanded by introducing a time-dependent factor. Experiments on benchmark datasets show that the bSB-based Ising solver offers superior performance in solution quality and convergence speed.