Abstract
We study an NP-hard problem motivated by energy-efficiently maintaining the connectivity of a symmetric wireless sensor communication network. Given an edge-weighted \(n\)-vertex graph, find a connected spanning subgraph of minimum cost, where the cost is determined by letting each vertex pay the most expensive edge incident to it in the subgraph. We provide an algorithm that works in polynomial time if one can find a set of obligatory edges that yield a spanning subgraph with \(O(\log n)\) connected components. We also provide a linear-time algorithm that reduces any input graph that consists of a tree together with \(g\) additional edges to an equivalent graph with \(O(g)\) vertices. Based on this, we obtain a polynomial-time algorithm for \(g\in O(\log n)\). On the negative side, we show that \(o(\log n)\)-approximating the difference \(d\) between the optimal solution cost and a natural lower bound is NP-hard and that there are presumably no exact algorithms running in \(2^{o(n)}\) time or in \(f(d)\cdot n^{O(1)}\) time for any computable function \(f\).
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Notes
- 1.
- 2.
To connect the \(c\) components of \(G_\ell \), one has to add \(c-1\) edges. These have at most \(2c-2\) end points. One can try all \(n^{2c-2}\) possibilities for choosing these end points and check each resulting graph for connectivity in \(O(n+m)\subseteq O(n^2)\) time.
- 3.
At most \(d\) vertices can pay more than their vertex lower bound. We can try all possibilities for choosing \(i\le d\) vertices, all \({d\atopwithdelims ()i}\) possibilities to increase their total cost by at most \(d\), and check whether the graph of the “paid” edges is connected. The algorithm runs in \(\sum _{i=1}^d{n\atopwithdelims ()i}{d\atopwithdelims ()i}\cdot O(n+m)\subseteq O(2^d\cdot n^{d+2})\) time.
References
Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)
Althaus, E., Călinescu, G., Mandoiu, I.I., Prasad, S.K., Tchervenski, N., Zelikovsky, A.: Power efficient range assignment for symmetric connectivity in static ad hoc wireless networks. Wirel. Netw. 12(3), 287–299 (2006)
Bentert, M., Fluschnik, T., Nichterlein, A., Niedermeier, R.: Parameterized aspects of triangle enumeration. In: Klasing, R., Zeitoun, M. (eds.) FCT 2017. LNCS, vol. 10472, pp. 96–110. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-55751-8_9
Betzler, N., van Bevern, R., Fellows, M.R., Komusiewicz, C., Niedermeier, R.: Parameterized algorithmics for finding connected motifs in biological networks. IEEE/ACM Trans. Comput. Biol. 8(5), 1296–1308 (2011)
Betzler, N., Guo, J., Komusiewicz, C., Niedermeier, R.: Average parameterization and partial kernelization for computing medians. J. Comput. Syst. Sci. 77(4), 774–789 (2011)
van Bevern, R., Komusiewicz, C., Sorge, M.: A parameterized approximation algorithm for the mixed and windy capacitated arc routing problem: theory and experiments. Networks (2017, in press)
Bruckner, S., Hüffner, F., Karp, R.M., Shamir, R., Sharan, R.: Topology-free querying of protein interaction networks. J. Comput. Biol. 17(3), 237–252 (2010)
Carmi, P., Katz, M.J.: Power assignment in radio networks with two power levels. Algorithmica 47(2), 183–201 (2007)
Clementi, A.E., Penna, P., Silvestri, R.: On the power assignment problem in radio networks. Mob. Netw. Appl. 9(2), 125–140 (2004)
Dost, B., Shlomi, T., Gupta, N., Ruppin, E., Bafna, V., Sharan, R.: Qnet: a tool for querying protein interaction networks. J. Comput. Biol. 15(7), 913–925 (2008)
Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, Heidelberg (2013). https://doi.org/10.1007/978-1-4471-5559-1
Erzin, A.I., Plotnikov, R.V., Shamardin, Y.V.: O nekotorykh polinomial’no razreshimykh sluchayakh i priblizhënnykh algoritmakh dlya zadachi postroyeniya optimal’nogo kommunikatsionnogo dereva. Diskretn. Anal. Issled. Oper. 20(1), 12–27 (2013)
Erzin, A.I., Mladenovic, N., Plotnikov, R.V.: Variable neighborhood search variants for min-power symmetric connectivity problem. Comput. Oper. Res. 78, 557–563 (2017)
Giacometti, A.: River networks. In: Complex Networks, Encyclopedia of Life Support Systems (EOLSS), pp. 155–180. EOLSS Publishers/UNESCO (2010)
Gutin, G., Wahlström, M., Yeo, A.: Rural postman parameterized by the number of components of required edges. J. Comput. Syst. Sci. 83(1), 121–131 (2017)
Hartung, S., Komusiewicz, C., Nichterlein, A.: Parameterized algorithmics and computational experiments for finding 2-clubs. J. Graph Algorithms Appl. 19(1), 155–190 (2015)
Hoffmann, S., Wanke, E.: Minimum power range assignment for symmetric connectivity in sensor networks with two power levels (2016). arXiv:1605.01752
Impagliazzo, R., Paturi, R.: On the complexity of \(k\)-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001)
Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)
Mertzios, G.B., Nichterlein, A., Niedermeier, R.: Linear-time algorithm for maximum-cardinality matching on cocomparability graphs. In: MFCS 2017. LIPIcs, vol. 83, pp. 46:1–46:14, Schloss Dagstuhl – Leibniz-Zentrum fuer Informatik (2017)
Montemanni, R., Gambardella, L.: Exact algorithms for the minimum power symmetric connectivity problem in wireless networks. Comput. Oper. Res. 32(11), 2891–2904 (2005)
Panigrahi, D.: Survivable network design problems in wireless networks. In: Proceedings of 22nd SODA, pp. 1014–1027. SIAM (2011)
Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proceedings of 29th STOC, pp. 475–484. ACM (1997)
Scott, J., Ideker, T., Karp, R.M., Sharan, R.: Efficient algorithms for detecting signaling pathways in protein interaction networks. J. Comput. Biol. 13(2), 133–144 (2006)
Sorge, M., van Bevern, R., Niedermeier, R., Weller, M.: From few components to an Eulerian graph by adding arcs. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 307–318. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25870-1_28
Sorge, M., van Bevern, R., Niedermeier, R., Weller, M.: A new view on rural postman based on Eulerian extension and matching. J. Discrete Alg. 16, 12–33 (2012)
Uhlmann, J., Weller, M.: Two-layer planarization parameterized by feedback edge set. Theoret. Comput. Sci. 494, 99–111 (2013)
Zalyubovskiy, V.V., Erzin, A.I., Astrakov, S.N., Choo, H.: Energy-efficient area coverage by sensors with adjustable ranges. Sensors 9(4), 2446–2460 (2009)
Zhang, H., Hou, J.C.: Maintaining sensing coverage and connectivity in large sensor networks. Ad Hoc Sens. Wirel. Netw. 1(1–2), 89–124 (2005)
Acknowledgments
RvB was supported by the Russian Science Foundation, grant 16-11-10041, while working on Sect. 2. The results in Sects. 3 and 4 were obtained during a research stay of RvB at TU Berlin, jointly supported by TU Berlin, by the Russian Foundation for Basic Research under grant 16-31-60007 mol\(\_\)a\(\_\)dk, and by the Ministry of Science and Education of the Russian Federation under the 5-100 Excellence Programme.
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Bentert, M., van Bevern, R., Nichterlein, A., Niedermeier, R. (2017). Parameterized Algorithms for Power-Efficient Connected Symmetric Wireless Sensor Networks. In: Fernández Anta, A., Jurdzinski, T., Mosteiro, M., Zhang, Y. (eds) Algorithms for Sensor Systems. ALGOSENSORS 2017. Lecture Notes in Computer Science(), vol 10718. Springer, Cham. https://doi.org/10.1007/978-3-319-72751-6_3
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