Abstract
We consider the Steiner tree problem in graphs under uncertainty, the so-called two-stage stochastic Steiner tree problem (SSTP). The problem consists of two stages: In the first stage, we do not know which nodes need to be connected. Instead, we know costs at which we may buy edges, and a set of possible scenarios one of which will arise in the second stage. Each scenario consists of its own terminal set, a probability, and second-stage edge costs. We want to find a selection of first-stage edges and second-stage edges for each scenario that minimizes the expected costs and satisfies all connectivity requirements. We show that SSTP is in the class of fixed-parameter tractable problems (FPT), parameterized by the number of terminals. Additionally, we transfer our results to the directed and the prize-collecting variant of SSTP.
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References
Bern, M.W., Plassmann, P.E.: The Steiner problem with edge lengths 1 and 2. Information Processing Letters 32(4), 171–176 (1989)
Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997)
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets Möbius: Fast subset convolution. In: STOC, pp. 67–74. ACM (2007)
Bomze, I., Chimani, M., Jünger, M., Ljubić, I., Mutzel, P., Zey, B.: Solving Two-Stage Stochastic Steiner Tree Problems by Two-Stage Branch-and-Cut. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010, Part I. LNCS, vol. 6506, pp. 427–439. Springer, Heidelberg (2010)
Byrka, J., Grandoni, F., Rothvoß, T., Sanitá, L.: An improved LP-based approximation for Steiner tree. In: STOC, pp. 583–592. ACM (2010)
Charikar, M., Chekuri, C., Cheung, T., Dai, Z., Goel, A., Guha, S., Li, M.: Approximation algorithms for directed Steiner problems. In: SODA, pp. 192–200. SIAM (1998)
Chimani, M., Mutzel, P., Zey, B.: Improved Steiner Tree Algorithms for Bounded Treewidth. In: Iliopoulos, C.S., Smyth, W.F. (eds.) IWOCA 2011. LNCS, vol. 7056, pp. 374–386. Springer, Heidelberg (2011)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer (1999)
Dreyfus, S.E., Wagner, R.A.: The Steiner problem in graphs. Networks 1(3), 195–207 (1971)
Fuchs, B., Kern, W., Mölle, D., Richter, S., Rossmanith, P., Wang, X.: Dynamic programming for minimum Steiner trees. Theory Computing Systems 41(3), 493–500 (2007)
Gupta, A., Hajiaghayi, M.T., Kumar, A.: Stochastic Steiner Tree with Non-uniform Inflation. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) APPROX/RANDOM 2007. LNCS, vol. 4627, pp. 134–148. Springer, Heidelberg (2007)
Gupta, A., Pál, M., Ravi, R., Sinha, A.: Boosted sampling: Approximation algorithms for stochastic optimization. In: STOC, pp. 417–426. ACM (2004)
Halperin, E., Krauthgamer, R.: Polylogarithmic inapproximability. In: STOC, pp. 585–594. ACM (2003)
Hwang, F., Richards, D., Winter, P.: The Steiner tree problem. Annals of discrete mathematics, vol. 53. North-Holland (1992)
Kahng, A.B., Robins, G.: On Optimal Interconnections for VLSI. Kluwer Academic Publishers (1995)
Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations, pp. 85–103. Plenum (1972)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Habilitation, Universität Tübingen (2002)
Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Dover Publications (1998)
Swamy, C., Shmoys, D.B.: Approximation Algorithms for 2-Stage Stochastic Optimization Problems. In: Arun-Kumar, S., Garg, N. (eds.) FSTTCS 2006. LNCS, vol. 4337, pp. 5–19. Springer, Heidelberg (2006)
Uchoa, E., de Aragão, M.P., Ribeiro, C.C.: Preprocessing Steiner problems from VLSI layout. Networks 40(1), 38–50 (2002)
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Kurz, D., Mutzel, P., Zey, B. (2013). Parameterized Algorithms for Stochastic Steiner Tree Problems. In: Kučera, A., Henzinger, T.A., Nešetřil, J., Vojnar, T., Antoš, D. (eds) Mathematical and Engineering Methods in Computer Science. MEMICS 2012. Lecture Notes in Computer Science, vol 7721. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36046-6_14
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DOI: https://doi.org/10.1007/978-3-642-36046-6_14
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