Advanced Quantum Annealing for the Bi-Objective Traveling Thief Problem: An $\varepsilon$-Constraint-based Approach

TL;DR

This paper introduces a hybrid quantum annealing approach combined with the $psilon$-constraint method to efficiently solve the complex bi-objective Traveling Thief Problem, demonstrating improved performance over traditional methods.

Contribution

It presents a novel reformulation of the BI-TTP into a QUBO model suitable for quantum annealing, integrating auxiliary variables and heuristics for enhanced solution quality.

Findings

Outperforms baseline methods in solution time

Effectively captures a broad Pareto front

Balances objectives with high solution diversity

Abstract

This paper addresses the Bi-Objective Traveling Thief Problem (BI-TTP), a challenging multi-objective optimization problem that requires the simultaneous optimization of travel cost and item profit. Conventional methods for the BI-TTP often face severe scalability issues due to the complex interdependence between routing and packing decisions, as well as the inherent complexity and large problem size. These difficulties render classical computing approaches increasingly inapplicable. To tackle this, we propose an advanced hybrid approach that combines quantum annealing (QA) with the $\varepsilon$-constraint method. Specifically, we reformulate the bi-objective problem into a single-objective formulation by restricting the second objective through adjustable $\varepsilon$-levels, determined within established upper and lower bounds. The resulting subproblem involves a sum of fractional terms, which is reformulated with auxiliary variables into an equivalent form. Subsequently, the equivalent formulation is transformed into a Quadratic Unconstrained Binary Optimization (QUBO) model, enabling direct solution via a quantum annealing (QA) solver. The solutions obtained from the quantum annealer are subsequently refined using a tailored heuristic procedure to further enhance overall performance. By leveraging the flexibility in selecting $\varepsilon$ parameters, our approach effectively captures a broad Pareto front, enhancing solution diversity. Experimental results on benchmark instances demonstrate that the proposed method effectively balances two objectives and outperforms baseline approaches in time efficiency.