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

Nguyen Hoang Viet, Nguyen Xuan Tung, Trinh Van Chien, Won-Joo Hwang

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.

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

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 -constraint method. Specifically, we reformulate the bi-objective problem into a single-objective formulation by restricting the second objective through adjustable -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 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.
Paper Structure (22 sections, 2 theorems, 36 equations, 5 figures, 3 tables, 3 algorithms)

This paper contains 22 sections, 2 theorems, 36 equations, 5 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Description of the BI-TTP
  4. QA-based $\varepsilon$-constraint method
  5. Preliminaries on Quantum Annealing
  6. Binary Reformulation of the BI-TTP
  7. Dispensing with $\varepsilon$-constraint method
  8. Finding solution to the BI-TTP
  9. Identification of Minimum and Maximum Knapsack Values
  10. The Travel Time Minimization with Constrained Knapsack's Values
  11. Update solution $(\mathbf{X}, \mathbf{Z})$ with fixed scalar $\mathbf{b}$
  12. Update auxiliary variable $\mathbf{b}$ based on received $(\mathbf{X}, \mathbf{Z})$
  13. Variable Utilization and Computational Complexity
  14. Numerical Results
  15. Experimental Setup
  16. ...and 7 more sections

Key Result

Lemma 1

Given the subproblem in prob: s-th segment subproblem - full, we represent it in a more concise form as follows: where $A_i\left(\mathbf{X}, \mathbf{Z}\right) = W\sum\nolimits_{u=1}^N\sum\nolimits_{v=1}^N d_{u,v} x_{u,i} x_{v,i+1}$ and $B_i \left(\mathbf{X}, \mathbf{Z}\right) = Wv_{\max} - W_i (v_{\max} - v_{\min}), \forall \, i = \overline{1,N}$. Here, $\mathcal{D}$ denotes the feasible domain d

Figures (5)

  • Figure 1: A four-city BI-TTP instance: To complete the tour in the minimum time, the thief can follow the route $c_1 \to c_3 \to c_2 \to c_4 \to c_1$ without collecting any items. Conversely, to maximize profit, the thief must collect items $\mathrm{item}_{2,1}$, $\mathrm{item}_{2,2}$, and $\mathrm{item}_{3,1}$, and travel along the route $c_1 \to c_4 \to c_2 \to c_3 \to c_1$ to complete the tour in the shortest possible time.
  • Figure 2: The HV of QA-based $\varepsilon$-constraint method, NSGA-II, U-NSGA-III, and MOEA/D at each iteration corresponding to the instance a280_n279. Our method produces a complete and high-quality Pareto front via Quantum Annealing, after which the evolving solution sets of NSGA-II, U-NSGA-III, and MOEA/D are incrementally compared at each iteration.
  • Figure 3: Comparison of Pareto fronts produced by NSGA-II, U-NSGA-III, MOEA/D, and the QA-based $\varepsilon$-constraint method for the instance a280_n279. Our approach consistently yields well-distributed solutions along the second objective across all instances, while NSGA-II and U-NSGA-III, show considerable variation across instances as a result of their reliance on the initialization process. Furthermore, MOEA/D fails to adequately explore the feasible region and frequently collapses to nearly identical values on the second objective.
  • Figure 4: The change in solution before and after applying LEA search
  • Figure 5: Sensitivity analysis of the QA-based $\varepsilon$-constraint method with respect to the number of segments $S$ and the initialization of the $\varepsilon$ levels.

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof
  • proof