Abstract
We study the problem of clearing connections by agents placed at some vertices in a directed graph. The agents can move only along directed paths. The objective is to minimize the number of agents guaranteeing that any pair of vertices can be connected by a underlying undirected path that can be cleared by the agents. We provide several results on the hardness, approximability and parameterized complexity of the problem. In particular, we show it to be: NP-hard, 2-approximable in polynomial-time, and solvable exactly in O(αn 322 α) time, where α is the number of agents in the solution. In addition, we give a simple linear-time algorithm optimally solving the problem in digraphs whose underlying graphs are trees. Finally, we discuss a related problem, where the task is to clear with a minimum number of agents a subgraph of the underlying graph containing its spanning tree. We show that this problem also admits a 2-approximation in polynomial time.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Alon, N., Prałat, P., Wormald, N.: Cleaning regular graphs with brushes. SIAM Journal on Discrete Mathematics 23(1), 233–250 (2008)
Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271 (1959)
Ford, L.R., Fulkerson, D.R.: Flows in Networks. Princeton University Press (1962)
Gaspers, S., Messinger, M.-E., Nowakowski, R.J., Prałat, P.: Clean the graph before you draw it! Information Processing Letters 109(10), 463–467 (2009)
Gaspers, S., Messinger, M.-E., Nowakowski, R.J., Prałat, P.: Parallel cleaning of a network with brushes. Discrete Applied Mathematics 158(5), 467–478 (2010)
Gordinowicz, P., Nowakowski, R.J., Prałat, P.: Polish — Let us play the cleaning game. Theoretical Computer Science 463, 123–132 (2012)
Held, M., Karp, R.M.: A dynamic programming approach to sequencing problems. Journal of the Society for Industrial and Applied Mathematics 10(1), 196–210 (1961)
Messinger, M.-E., Prałat, P., Nowakowski, R.J., Wormald, N.: Cleaning random d-regular graphs with brushes using a degree-greedy algorithm. In: Janssen, J., Prałat, P. (eds.) CAAN 2007. LNCS, vol. 4852, pp. 13–26. Springer, Heidelberg (2007)
Messinger, M.-E., Nowakowski, R.J., Prałat, P.: Cleaning a network with brushes. Theoretical Computer Science 399, 191–205 (2008)
Messinger, M.-E., Nowakowski, R.J., Prałat, P.: Cleaning with Brooms. Graphs and Combinatorics 27(2), 251–267 (2011)
Prałat, P.: Cleaning random graphs with brushes. Australasian Journal of Combinatorics 43, 237–251 (2009)
Prałat, P.: Cleaning random d-regular graphs with Brooms. Graphs and Combinatorics 27(4), 567–584 (2011)
Ren, W., Zhao, Q.: A note on Algorithms for connected set cover problem and fault-tolerant connected set cover problem. Theoretical Computer Science 412(45), 6451–6454 (2011)
Shuai, T.-P., Hu, X.-D.: Connected set cover problem and its applications. In: Cheng, S.-W., Poon, C.K. (eds.) AAIM 2006. LNCS, vol. 4041, pp. 243–254. Springer, Heidelberg (2006)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Levcopoulos, C., Lingas, A., Nilsson, B.J., Żyliński, P. (2014). Clearing Connections by Few Agents. In: Ferro, A., Luccio, F., Widmayer, P. (eds) Fun with Algorithms. FUN 2014. Lecture Notes in Computer Science, vol 8496. Springer, Cham. https://doi.org/10.1007/978-3-319-07890-8_25
Download citation
DOI: https://doi.org/10.1007/978-3-319-07890-8_25
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-07889-2
Online ISBN: 978-3-319-07890-8
eBook Packages: Computer ScienceComputer Science (R0)