Efficient Bit Labeling in Factorization Machines with Annealing for Traveling Salesman Problem

TL;DR

The paper tackles the challenge of efficiently solving large-scale combinatorial optimization problems, specifically the Traveling Salesman Problem (TSP), by formulating a QUBO surrogate via Factorization Machines with annealing (FMA). It introduces Gray labeling, based on inversion numbers and Gray coding, as an alternative to natural labeling to map routes to binary bitstrings, with lengths $\ell_N = \lceil \log (N-1)! \rceil$ and $\ell_G = \sum_{i=2}^{N-1} \lceil \log i \rceil$, and defines forward and inverse mappings for both schemes. A local-solution metric $p = \mathbb{E}_{\mathbf{b}}[f(\underline{\mathbf{b}})]$ is proposed to quantify the prevalence of local minima, showing Gray labeling reduces this risk as $N$ grows. Numerical simulations across $N=5$–$15$ demonstrate Gray labeling achieving smaller $d_{\min}$ and faster convergence, with global optima found at several sizes and still competitive at larger sizes, suggesting Gray labeling improves the practical performance of QUBO/FM solvers for routing problems. Together, the Gray labeling framework and the local-solution metric offer actionable guidance for configuring annealing-based optimizers in binary encodings of combinatorial problems.

Abstract

To efficiently find an optimum parameter combination in a large-scale problem, it is a key to convert the parameters into available variables in actual machines. Specifically, quadratic unconstrained binary optimization problems are solved with the help of machine learning, e.g., factorization machines with annealing, which convert a raw parameter to binary variables. This work investigates the dependence of the convergence speed and the accuracy on binary labeling method, which can influence the cost function shape and thus the probability of being captured at a local minimum solution. By exemplifying traveling salesman problem, we propose and evaluate Gray labeling, which correlates the Hamming distance in binary labels with the traveling distance. Through numerical simulation of traveling salesman problem up to 15 cities at a limited number of iterations, the Gray labeling shows less local minima percentages and shorter traveling distances compared with natural labeling.

Figures (3)

• Figure 1: The local solution metric $p$ for two labeling methods as a function of the number of cities $N$.
• Figure 2: Numerically obtained shortest distance $d_{\mathrm{min}}$ until the step normalized by the optimum one in FMA-based TSP for (a) $N =$ 5 to 9 and (b) $N =$ 11 to 15.
• Figure 3: Numerically obtained results of TSP routes in 13 cities: routes obtained by (a) natural labeling and (b) Gray labeling compared with (c) the optimal route, respectively.