Parameterized Directed \(k\)-Chinese Postman Problem and \(k\) Arc-Disjoint Cycles Problem on Euler Digraphs
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, 124–128 (2013)) 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-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-hard even for Euler digraphs.
Research of GG was supported by Royal Society Wolfson Research Merit Award.
- 2.van Bevern, R., Niedermeier, R., Sorge, M., Weller, M.: Complexity of arc routing problems. In: Corberán, A., Laporte, G. (eds.) Arc Routing: Problems, Methods and Applications, SIAM, Phil. (in press)Google Scholar
- 10.Lucchesi, C.L.: A minimax equality for directed graphs. Ph.D. thesis, University of Waterloo, Ontario, Canada (1976)Google Scholar
- 15.Sorge, M.: Some Algorithmic Challenges in Arc Routing. Talk at NII Shonan Seminar no. 18, May 2013Google Scholar
- 20.Zhang, L.: Polynomial algorithms for the \(k\)-Chinese postman problem. In: Information Processing ’92, vol. 1, pp. 430–435 (1992)Google Scholar