Approximation Schemes for Orienteering and Deadline Traveling Salesman Problem in Doubling Metrics

Khái niệm cốt lõi

This paper presents the first approximation schemes for the deadline Traveling Salesman Problem (TSP) on metrics with bounded doubling dimension, which includes Euclidean metrics of fixed dimension. The authors also provide quasi-polynomial time approximation schemes for the k-stroll and point-to-point (P2P) orienteering problems on such metrics.

Tóm tắt

The paper studies fundamental variants of the Traveling Salesman Problem (TSP) in which the goal is to visit as many customers as possible before their required service deadlines.
The key results are:

For graphs with bounded treewidth, the authors present an exact algorithm for solving the k-stroll and P2P orienteering problems, and use this to obtain a quasi-polynomial time approximation scheme (QPTAS) for deadline TSP on such graphs when the distances and deadlines are integers.

For metrics with bounded doubling dimension (which includes Euclidean metrics of fixed dimension):

They provide a QPTAS for the k-stroll problem.
They use this to obtain a QPTAS for the P2P orienteering problem.
Building on these results, they present the first approximation scheme for the deadline TSP problem on such metrics, when the distances and deadlines are integers.
The algorithm can also be adapted to obtain a bicriteria (1+ε, 1+ε)-approximation for deadline TSP when the distances (and deadlines) are in Q+.

The key technical ideas include:

Exploiting the structure of bounded treewidth graphs to design dynamic programming algorithms.
Leveraging the hierarchical decomposition of doubling metrics to obtain quasi-polynomial time approximation schemes.
Carefully bounding the excess of paths to enable approximation schemes for deadline TSP.

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Approximation Schemes for Orienteering and Deadline TSP in Doubling Metrics

by Kinter Ren,M... lúc arxiv.org 05-03-2024

https://arxiv.org/pdf/2405.00818.pdf

Yêu cầu sâu hơn

How can the techniques developed in this paper be extended to other variants of vehicle routing problems, such as capacitated vehicle routing or TSP with neighborhoods, on doubling metrics

The techniques developed in this paper for solving Orienteering and Deadline TSP problems on graphs with bounded treewidth and doubling metrics can be extended to other variants of vehicle routing problems on doubling metrics, such as capacitated vehicle routing or TSP with neighborhoods, by adapting the dynamic programming approach and hierarchical decomposition used in this paper.
For capacitated vehicle routing, the capacity constraints can be incorporated into the DP algorithm by considering the capacity of each vehicle and the demand at each node. The hierarchical decomposition can help in breaking down the problem into smaller subproblems based on the capacity constraints and the demand at each node, similar to how it was used for Orienteering and Deadline TSP.
Similarly, for TSP with neighborhoods, the neighborhoods of each node can be defined and considered in the DP algorithm to ensure that the path visits nodes within their respective neighborhoods. The hierarchical decomposition can assist in structuring the neighborhoods and optimizing the path within these constraints.
Overall, by customizing the DP algorithm and utilizing the hierarchical decomposition technique, the methods developed in this paper can be applied to various variants of vehicle routing problems on doubling metrics.

Are there any hardness of approximation results known for deadline TSP on doubling metrics or other non-trivial metric spaces

There are hardness of approximation results known for deadline TSP on doubling metrics and other non-trivial metric spaces. While this paper focuses on providing approximation schemes for Orienteering and Deadline TSP on graphs with bounded doubling dimension and treewidth, the complexity of approximating deadline TSP on general metrics remains a challenging problem.
In the context of doubling metrics, the hardness of approximation results for deadline TSP may vary based on the specific characteristics of the metric space. For example, on Euclidean metrics with fixed dimensions, the complexity of approximating deadline TSP may differ from that on general doubling metrics.
Further research and analysis are required to explore the hardness of approximation results for deadline TSP on doubling metrics and other non-trivial metric spaces, taking into account the specific properties and constraints of each metric space.

Can the approximation schemes presented in this paper be made more practical and efficient, perhaps by exploiting additional structural properties of the underlying graphs or metrics

To make the approximation schemes presented in this paper more practical and efficient, additional structural properties of the underlying graphs or metrics can be exploited. Here are some ways to enhance the practicality and efficiency of the approximation schemes:

Graph Reduction Techniques: Utilize graph reduction techniques to simplify the input graph while preserving the essential characteristics relevant to the vehicle routing problem. This can help in reducing the computational complexity of the algorithms.

Preprocessing Steps: Implement preprocessing steps to identify and eliminate redundant nodes or edges in the graph. This can streamline the computation and improve the efficiency of the approximation schemes.

Heuristic Approaches: Incorporate heuristic approaches to guide the search for optimal or near-optimal solutions. Heuristics can help in quickly identifying promising paths and reducing the search space.

Parallelization: Explore parallel computing techniques to distribute the computational load across multiple processors or cores. This can speed up the computation of the approximation schemes.

Fine-tuning Parameters: Fine-tune the parameters of the approximation algorithms based on the specific characteristics of the input instances. Adjusting parameters like epsilon values or thresholds can improve the accuracy and efficiency of the solutions.

By implementing these strategies and leveraging additional structural properties of the graphs or metrics, the approximation schemes presented in this paper can be made more practical and efficient for real-world applications.