Parameterized Directed $k$-Chinese Postman Problem and $k$ Arc-Disjoint Cycles Problem on Euler Digraphs

TL;DR

This paper proves that the parameterized Directed $k$-Chinese Postman Problem and the $k$ arc-disjoint cycles problem on Euler digraphs are fixed-parameter tractable, extending understanding of their computational complexity.

Contribution

It establishes fixed-parameter tractability for these problems on Euler digraphs, addressing open questions about their complexity.

Findings

Proves $k$-DCPP is fixed-parameter tractable.

Shows $k$ arc-disjoint cycles problem is FPT on Euler digraphs.

Vertex-disjoint cycles problem remains W[1]-hard on Euler digraphs.

Abstract

In the Directed $k$-Chinese Postman Problem ($k$-DCPP), we are given a connected weighted digraph $G$ and asked to find $k$ non-empty closed directed walks covering all arcs of $G$ such that the total weight of the walks is minimum. Gutin, Muciaccia and Yeo (Theor. Comput. Sci. 513 (2013) 124--128) asked for the parameterized complexity of $k$-DCPP when $k$ is the parameter. We prove that the $k$-DCPP is fixed-parameter tractable. We also consider a related problem of finding $k$ arc-disjoint directed cycles in an Euler digraph, parameterized by $k$. Slivkins (ESA 2003) showed that this problem is W[1]-hard for general digraphs. Generalizing another result by Slivkins, we prove that the problem is fixed-parameter tractable for Euler digraphs. The corresponding problem on vertex-disjoint cycles in Euler digraphs remains W[1]-hard even for Euler digraphs.