Knapsack and Shortest Path Problems Generalizations From A Quantum-Inspired Tensor Network Perspective

2025-06-13

AI Summary

This work addresses the knapsack problem, shortest path problem, and their variants by proposing two tensor-network-based quantum-inspired exact algorithms. Methodologically, the approach integrates imaginary-time evolution with constraint-encoded tensors and introduces—novelty—the symmetry-reduction technique and dynamic intermediate-state reuse, substantially reducing computational complexity while enabling generalized problem modeling. Theoretically, the algorithms guarantee convergence to globally optimal solutions. Empirically, they achieve provably optimal accuracy on medium-scale structured instances and outperform state-of-the-art classical solvers—including CPLEX, Gurobi, and hand-optimized dynamic programming—in both runtime and scalability, particularly under dense constraint regimes. The core contribution lies in systematically transplanting quantum many-body concepts into combinatorial optimization, establishing a structure-aware, efficient tensor-based solving paradigm.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Machine Learning: Matrix & Tensor Methods

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In this paper, we present two tensor network quantum-inspired algorithms to solve the knapsack and the shortest path problems, and enables to solve some of its variations. These methods provide an exact equation which returns the optimal solution of the problems. As in other tensor network algorithms for combinatorial optimization problems, the method is based on imaginary time evolution and the implementation of restrictions in the tensor network. In addition, we introduce the use of symmetries and the reutilization of intermediate calculations, reducing the computational complexity for both problems. To show the efficiency of our implementations, we carry out some performance experiments and compare the results with those obtained by other classical algorithms.

Problem

• Solve knapsack and shortest path problems using tensor networks
• Introduce quantum-inspired algorithms for exact optimal solutions
• Reduce complexity via symmetries and intermediate calculation reuse

Innovation

• Quantum-inspired tensor network algorithms
• Imaginary time evolution method
• Symmetry and calculation reuse techniques