## Abstract

The Secluded Path problem models a situation where sensitive information has to be transmitted between a pair of nodes along a path in a network. The measure of the quality of a selected path is its *exposure cost*, which is the total cost of vertices in its closed neighborhood. The task is to select a *secluded* path, i.e., a path with a small exposure cost. Similarly, the Secluded Steiner Tree problem is to find a tree in a graph connecting a given set of terminals such that the exposure cost of the tree is minimized. In this paper we present a systematic study of the parameterized complexity of Secluded Steiner Tree. In particular, we establish the tractability of Secluded Path being parameterized by “above guarantee” value, which in this case is the length of a shortest path between vertices. We also show how to extend this result for Secluded Steiner Tree, in this case we parameterize above the size of an optimal Steiner tree and the number of terminals. We also consider various parameterization of the problems such as by the treewidth, the size of a vertex cover, feedback vertex set, or the maximum vertex degree and establish kernelization complexity of the problem subject to different choices of parameters.

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## Notes

Gutin et al. prove in [21] the statement for the dual Hitting Set problem.

## References

Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM

**42**, 844–856 (1995)Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. Discret. Math.

**12**, 289–297 (1999)Becker, A., Geiger, D.: Optimization of Pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artif. Intell.

**83**, 167–188 (1996)Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets Möbius: fast subset convolution. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pp 67–74. ACM, California, USA (2007)

Bodlaender, H. L., Downey, R. G., Fellows, M. R., Hermelin, D.: On problems without polynomial kernels. J. Comput. Syst. Sci.

**75**, 423–434 (2009)Bodlaender, H. L., Jansen, B. M. P., Kratsch, S.: Kernelization lower bounds by cross-composition. SIAM J. Discret. Math.

**28**, 277–305 (2014)Cai, L.: Parameterized complexity of cardinality constrained optimization problems. Comput. J.

**51**, 102–121 (2008)Cai, L., Chan, S. M., Chan, S. O.: Random separation: a new method for solving fixed-cardinality optimization problems. In: IWPEC, vol. 4169 of Lecture Notes in Computer Science, Springer, pp 239–250 (2006)

Chechik, S., Johnson, M. P., Parter, M., Peleg, D.: Secluded connectivity problems. CoRR, abs/1212, 6176 (2012)

Chechik, S., Johnson, M. P., Parter, M., Peleg, D.: Secluded connectivity problems. In: Proceedings of the 21st Annual European Symposium Algorithms (ESA), vol. 8125 of Lecture Notes in Computer Science, pp 301–312. Springer (2013)

Cygan, M., Fomin, F. V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized algorithms. Springer (2015)

Downey, R. G., Fellows, M. R.: Fundamentals of parameterized complexity. Texts in Computer Science, Springer (2013)

Dreyfus, S. E., Wagner, R. A.: The Steiner problem in graphs. Networks

**1**, 195–207 (1971)Fellows, M. R., Hermelin, D., Rosamond, F. A., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci.

**410**, 53–61 (2009)Fomin, F. V., Golovach, P. A., Karpov, N., Kulikov, A.S.: Parameterized complexity of secluded connectivity problems. In: FSTTCS 2015, vol. 45 of LIPIcs, pp 408–419

Fomin, F. V., Kratsch, D.: Exact exponential algorithms, Texts in Theoretical Computer Science. An EATCS Series, Springer-Verlag (2010)

Fomin, F. V., Villanger, Y.: Treewidth computation and extremal combinatorics. Combinatorica

**32**, 289–308 (2012)Gao, J., Zhao, Q., Swami, A.: The thinnest path problem for secure communications: A directed hypergraph approach. In: Proceedings of the 50th Annual Allerton Conference on Communication, Control, and Computing, pp 847–852. IEEE

Garey, M. R., Johnson, D. S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman (1979)

Gilbers, A.: Visibility domains and complexity. PhD thesis, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn (2013)

Gutin, G., Jones, M., Yeo, A.: Kernels for below-upper-bound parameterizations of the hitting set and directed dominating set problems. Theor. Comput. Sci.

**412**, 5744–5751 (2011)Johnson, M. P., Liu, O., Rabanca, G.: Secluded path via shortest path. In: SIROCCO 2014, vol. 8576 of Lecture Notes in Computer Science, pp 108–120. Springer (2014)

Karp, R. M.: Reducibility among combinatorial problems. In: Proceedings of a symposium on the Complexity of Computer Computations, The IBM Research Symposia Series, pp 85–103. Plenum Press, New York (1972)

Naor, M., Schulman, L., Srinivasan, A.: Splitters and nearoptimal derandomization. In: 36th Annual Symposium on Foundations of Computer Science (FOCS 1995), pp 182–191. IEEE (1995)

Nederlof, J.: Fast polynomial-space algorithms using inclusion-exclusion. Algorithmica

**65**, 868–884 (2013)Pietrzak, K.: On the parameterized complexity of the fixed alphabet shortest common supersequence and longest common subsequence problems. J. Comput. Syst. Sci.

**67**, 757–771 (2003)

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A preliminary version of this paper appeared as an extended abstract in the proceedings of FSTTCS 2015 [15]. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 267959 and the Government of the Russian Federation (grant 14.Z50.31.0030).

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Fomin, F.V., Golovach, P.A., Karpov, N. *et al.* Parameterized Complexity of Secluded Connectivity Problems.
*Theory Comput Syst* **61**, 795–819 (2017). https://doi.org/10.1007/s00224-016-9717-x

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DOI: https://doi.org/10.1007/s00224-016-9717-x