The Hierarchical Chinese Postman Problem: the slightest disorder makes it hard, yet disconnectedness is manageable

TL;DR

This paper studies the complexity and approximation algorithms for the Hierarchical Chinese Postman Problem, revealing NP-hardness even in simplified cases and providing approximation solutions for linearly ordered classes.

Contribution

It demonstrates NP-hardness for connected classes and offers 5/3-approximation algorithms for linearly ordered classes, extending Rural Postman Problem results.

Findings

NP-hardness for connected classes even with simple order structures

5/3-approximation algorithms for linearly ordered classes

Transfer of Rural Postman Problem techniques to the hierarchical setting

Abstract

The Hierarchical Chinese Postman Problem is finding a shortest traversal of all edges of a graph respecting precedence constraints given by a partial order on classes of edges. We show that the special case with connected classes is NP-hard even on orders decomposable into a chain and an incomparable class. For the case with linearly ordered (possibly disconnected) classes, we get 5/3-approximations and fixed-parameter algorithms by transferring results from the Rural Postman Problem.