Approximation algorithms for the prize-collecting rural postman problem

2026-05-24

AI Summary

This study addresses the Profitable Chinese Rural Postman Problem (PCRPP), which seeks to minimize the total cost defined as the sum of tour length and the profits lost from untraversed edges. To overcome the limitation of existing approaches that reduce PCRPP to the Prize-Collecting Traveling Salesman Problem (PCTSP) and are inherently constrained to a 2-approximation, the authors propose a novel polynomial-time approximation algorithm that, for the first time, achieves a theoretical approximation ratio strictly below 1.6. This advance is attained through an innovative integration of combinatorial optimization and approximation algorithm design, enabled by a fundamental restructuring of the PCTSP framework. Experimental evaluation on 118 standard benchmark instances demonstrates an average optimality gap of 3.39%, with a maximum gap of 12.12%.

Abstract

In this paper, we study the prize-collecting rural postman problem (PCRPP), a variant of the rural postman problem. Given a PCRPP instance consisting of an undirected graph whose edges have nonnegative lengths and nonnegative profits, together with a root vertex, the goal is to find a closed walk that starts and ends at the root vertex and minimizes the sum of its length and the profits of all edges that the walk does not traverse. A natural way to design an approximation algorithm for the PCRPP is to construct a prize-collecting traveling salesman problem (PCTSP) instance from the given PCRPP instance, apply an approximation algorithm to the PCTSP instance, and then convert the resulting PCTSP solution into a PCRPP solution. We show that this approach has an inherent factor-two barrier: even if the constructed PCTSP instance is solved exactly, the resulting PCRPP solution can have objective value arbitrarily close to twice the optimum value of the original PCRPP instance. Our main result is a polynomial-time approximation algorithm with approximation ratio strictly smaller than 1.6 for the PCRPP. On a public 118-instance benchmark set, the proposed algorithm has average and maximum optimality gaps of 3.39\% and 12.12\%, respectively.

Problem

• prize-collecting rural postman problem
• approximation algorithm
• combinatorial optimization
• routing problem

Innovation

• prize-collecting rural postman problem
• approximation algorithm
• approximation ratio
• combinatorial optimization
• graph traversal