Parameterized Complexity of k-Chinese Postman Problem

TL;DR

This paper proves that the k-Chinese Postman Problem is fixed-parameter tractable and admits a kernel with O(k^2 log k) vertices, advancing understanding of its computational complexity.

Contribution

It establishes fixed-parameter tractability and kernelization results for the k-Chinese Postman Problem, answering open questions about its complexity.

Findings

k-CPP is fixed-parameter tractable

Existence of a kernel with O(k^2 log k) vertices

Directed k-CPP remains NP-complete

Abstract

We consider the following problem called the $k$-Chinese Postman Problem ($k$-CPP): given a connected edge-weighted graph $G$ and integers $p$ and $k$, decide whether there are at least $k$ closed walks such that every edge of $G$ is contained in at least one of them and the total weight of the edges in the walks is at most $p$? The problem $k$-CPP is NP-complete, and van Bevern et al. (to appear) and Sorge (2013) asked whether the $k$-CPP is fixed-parameter tractable when parameterized by $k$. We prove that the $k$-CPP is indeed fixed-parameter tractable. In fact, we prove a stronger result: the problem admits a kernel with $O(k^2\log k)$ vertices. We prove that the directed version of $k$-CPP is NP-complete and ask whether the directed version is fixed-parameter tractable when parameterized by $k$.