Efficient Operator Design for Feasible Solution Mapping in Pickup-and-Delivery Traveling Salesman Problem

Core Concepts

This paper proposes a unified set of operators that consistently map one feasible PDTSP solution to another, enabling efficient exploration of the feasible solution space without wasting time on infeasible solutions.

Abstract

The paper addresses the pickup-and-delivery traveling salesman problem (PDTSP), which requires finding the shortest tour along a sequence of one-to-one pickup-and-delivery nodes while satisfying precedence constraints. Classic operations research algorithms for PDTSP struggle to scale to large problem sizes.
The key contributions are:

A unified operator set containing various operators that map one feasible PDTSP tour to another, without violating precedence constraints. The operators include intra-block node-exchange, inter-block node-exchange, node pair-exchange, same-type block-exchange, and mixed-type block-exchange.

Utilizing the unified operator set within a reinforcement learning (RL) framework to solve PDTSP, where operators are evaluated and selected as policies to efficiently explore the feasible solution space.

Experimental results demonstrating the effectiveness of the proposed method, which outperforms classic operations research algorithms and existing learning-based approaches in terms of solution quality and computational efficiency, especially for large-scale PDTSP instances.

Stats

The total number of Hamiltonian cycles in PDTSP is (2n)!, while the number of feasible Hamiltonian cycles (i.e., tours) is (2n)!/2n.
As the problem size increases, the feasible solutions account for a diminishing portion of the entire solution space.

Quotes

"To tackle such a challenge, this paper leverages learning operators that consistently map one feasible solution to another."
"We propose a unified operator set containing various operators that map one feasible tour to another by confining the solution search within a feasible space."

Key Insights Distilled From

Learn to Tour: Operator Design For Solution Feasibility Mapping in Pickup-and-delivery Traveling Salesman Problem

by Bowen Fang,X... at arxiv.org 04-18-2024

https://arxiv.org/pdf/2404.11458.pdf

Deeper Inquiries

How can the proposed operator-based RL framework be extended to solve other variants of the traveling salesman problem, such as the multi-vehicle or time-window constrained versions

The proposed operator-based RL framework can be extended to solve other variants of the traveling salesman problem by adapting the operators and the state representation to accommodate the specific constraints of each variant.
For the multi-vehicle version of the traveling salesman problem, the operators can be modified to handle the assignment of nodes to different vehicles and the sequencing of visits for each vehicle. The state representation would need to include information about the current assignments of nodes to vehicles and the order of visits for each vehicle. By incorporating these changes, the RL framework can learn to optimize the routes for multiple vehicles simultaneously.
In the time-window constrained version of the traveling salesman problem, the operators can be designed to respect the time constraints at each node. This could involve operators that rearrange the sequence of visits to ensure that all nodes are visited within their specified time windows. The state representation would include information about the time windows for each node and the current time of the tour, allowing the RL framework to make decisions that respect these constraints.
By adapting the operators and state representation to address the specific requirements of each variant, the operator-based RL framework can be effectively applied to solve a wide range of traveling salesman problem variations.

What are the potential limitations of the current operator design, and how could they be addressed to further improve the solution quality and computational efficiency

One potential limitation of the current operator design is that the operators may not be able to effectively handle complex problem instances with a large number of nodes or intricate constraints. To address this limitation and further improve solution quality and computational efficiency, several strategies can be implemented:

Advanced Operator Design: Develop more sophisticated operators that can handle complex constraints and interactions between nodes more effectively. This could involve designing operators that consider multiple nodes simultaneously or incorporate domain-specific knowledge to guide the search process.

Dynamic Operator Selection: Implement a mechanism for dynamically selecting operators based on the current state of the tour. By adapting the choice of operators to the specific characteristics of the problem instance, the RL framework can explore the solution space more efficiently.

Ensemble of Operators: Combine multiple operators into an ensemble approach to leverage the strengths of each operator. By using a diverse set of operators, the framework can explore a wider range of solutions and potentially find better-quality tours.

Fine-tuning Hyperparameters: Experiment with different hyperparameters, such as learning rates and exploration strategies, to optimize the performance of the RL framework. Fine-tuning these parameters can help improve convergence speed and solution quality.

By addressing these potential limitations and implementing these strategies, the operator-based RL framework can enhance its capabilities in solving complex variants of the traveling salesman problem.

Given the insights from this work, how could the principles of feasible solution mapping be applied to other combinatorial optimization problems beyond the traveling salesman problem

The principles of feasible solution mapping demonstrated in this work can be applied to other combinatorial optimization problems beyond the traveling salesman problem. By focusing on operators that consistently map one feasible solution to another, the framework can effectively explore the solution space without wasting computational resources on infeasible solutions.
For example, in the vehicle routing problem, where multiple vehicles need to visit a set of locations while respecting capacity and time constraints, feasible solution mapping can ensure that the routes generated adhere to these constraints. By designing operators that consider the specific requirements of the vehicle routing problem, such as vehicle capacity and time windows, the framework can efficiently find high-quality solutions.
Similarly, in the job scheduling problem, where tasks need to be assigned to machines while minimizing completion time or maximizing resource utilization, feasible solution mapping can guide the search process towards feasible schedules. Operators that respect task dependencies and machine availability can help generate optimal schedules that meet all constraints.
Overall, the principles of feasible solution mapping can be a valuable approach in various combinatorial optimization problems, ensuring that the generated solutions are valid and of high quality.