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Constrained parallel tempering in traveling-salesman problems with circular neighborhoods

Andrea Grimaldi1,2, Davi R. Rodrigues1, Eleonora Raimondo3, Vito Puliafito1, Vincenza Crupi4, Vitoantonio Bevilacqua1, Mario Carpentieri1, Francesca Garesci2, and Giovanni Finocchio3,4,*

  • *Contact author: gfinocchio@unime.it
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Phys. Rev. Applied 25, 024023 – Published 6 February, 2026

DOI: https://doi.org/10.1103/plr8-xys1

Abstract

This paper is a contribution to the Physical Review Applied collection titled Physics-Inspired Computing.

Combinatorial optimization problems such as the traveling-salesman problem (TSP) are central to many applications ranging from physics to mechanical engineering, such as toolpath planning, robotic navigation, and drone routing. In this work, we propose the concept of constrained parallel tempering (CPT), an energy-minimization algorithm implemented on probabilistic Ising machines (PIMs) with probabilistic bits (p-bits) tailored for constrained problems. CPT separates the constraint and target components of the encoding into distinct energy terms with independent sampling temperatures, improving convergence and reducing computational overhead compared with traditional parallel tempering. We demonstrate the effectiveness of CPT on a benchmark of a TSP instance and extend the method to a more general formulation, the TSP with circular neighborhoods (TSPCNs), where each node is defined as a region, not a point. This model reflects real-world flexibility in robotic and drone-based applications. A hybrid deterministic–Monte Carlo refinement is then used to handle the continuous solution space of TSPCNs. Additionally, we present a hardware-friendly approach to generate Gaussian-distributed random numbers using a PIM with p-bits. Our results suggest that CPT-PIM frameworks offer an approach with better scalability and potentially low-power implementation for embedding intelligent decision-making into edge computing applications.

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This article appears in the following collection:

Collection on Physics-Inspired Computing

Physical Review Applied is pleased to present a Collection on Physics-Inspired Computing, highlighting the rapidly evolving field of energy-efficient computing techniques, from hardware technologies to algorithms, where physics inspiration serves as the crucial link. Contributions to this collection will be published throughout 2025. This Collection is being curated by Guest Editors Kerem Camsari and Supriyo Datta.

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Outline

Information

Phys. Rev. Applied 25, 024023– Published 6 February, 2026
  • Received 24 June 2025
  • Revised 15 November 2025
  • Accepted 6 January 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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