Abstract
We consider the problem of finding shortest paths in a graph with independent randomly distributed edge lengths. Our goal is to maximize the probability that the path length does not exceed a given threshold value (deadline). We give a surprising exact n Θ(logn) algorithm for the case of normally distributed edge lengths, which is based on quasi-convex maximization. We then prove average and smoothed polynomial bounds for this algorithm, which also translate to average and smoothed bounds for the parametric shortest path problem, and extend to a more general non-convex optimization setting. We also consider a number other edge length distributions, giving a range of exact and approximation schemes.
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.
References
Ackermann, H., Newman, A., Röglin, H., Vöcking, B.: Decision making based on approximate and smoothed pareto curves. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 675–684. Springer, Heidelberg (2005)
Bertsekas, D.: Dynamic Programming and Optimal Control, 2nd edn., vol. II. Athena Scientific, Belmont (2001)
Boyan, J., Mitzenmacher, M.: Improved results for route planning in stochastic transportation networks. In: Proc. of Symposium of Discrete Algorithms (2001)
Carstensen, P.: The complexity of some problems in parametric linear and combinatorial programming. Ph.D. Thesis, Mathematics Dept., U. of Michigan, Ann Arbor, Mich. (1983)
Dean, B., Goemans, M., Vondrak, J.: Approximating the stochastic knapsack: The benefit of adaptivity. In: Proceedings of FOCS, pp. 208–217 (2004)
Fan, Y., Kalaba, R., Moore, I.J.E.: Arriving on time. Journal of Optimization Theory and Applications (forthcoming)
Goel, A., Indyk, P.: Stochastic load balancing and related problems. In: Proceedings of the 40th Symposium on Foundations of Computer Science (1999)
Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization. Kluwer Academic Publishers, Dordrecht (2000)
Karger, D., Motwani, R., Ramkumar, G.D.S.: On approximating the longest path in a graph. Algorithmica 18, 82–98 (1997)
Kelner, J.A., Spielman, D.A.: A randomized polynomial-time simplex algorithm for linear programming. Electronic Colloquium on Computational Complexity (156) (2005)
Kleinberg, J., Rabani, Y., Tardos, É.: Allocating bandwidth for bursty connections. SIAM Journal on Computing 30(1), 191–217 (2000)
Loui, R.P.: Optimal paths in graphs with stochastic or multidimentional weights. Communications of the ACM 26, 670–676 (1983)
Miller-Hooks, E.D., Mahmassani, H.S.: Least expected time paths in stochastic, time-varying transportation networks. Transportation Science 34, 198–215 (2000)
Mirchandani, P., Soroush, H.: Optimal paths in probabilistic networks: A case with temporary preferences. Computers and Operations Research 12(4), 365–381 (1985)
Mote, J., Murthy, I., Olson, D.: A parametric approach to solving bicriterion shortest path problems. European Journal of Operational Research 53, 81–92 (1991)
Mulmuley, K., Shah, P.: A lower bound for the shortest path problem. Journal of Computer and System Sciences 63(2), 253–267 (2001)
Nikolova, E., Brand, M., Karger, D.R.: Optimal route planning under uncertainty. In: Proceedings of International Conference on Automated Planning and Scheduling (2006)
Nikolova, E., Kelner, J.A.: On the hardness and smoothed complexity of low-rank quasi-concave minimization (May 2006) (manuscript)
Pallottino, S., Scutella, M.G.: Shortest path algorithms in transportation models: Classical and innovative aspects. Technical Report TR-97-06, Universita di Pisa Dipartimento di Informatica, Pisa, Italy (1997)
Papadimitriou, C.H., Yannakakis, M.: Shortest paths without a map. Theoretical Computer Science 84, 127–150 (1991)
Papadimitriou, C.H., Yannakakis, M.: On limited nondeterminism and the complexity of the V-C dimension. Journal of Computer and System Sciences 53(2), 161–170 (1996)
Polychronopoulos, G.H., Tsitsiklis, J.N.: Stochastic shortest path problems with recourse. Networks 27(2), 133–143 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Nikolova, E., Kelner, J.A., Brand, M., Mitzenmacher, M. (2006). Stochastic Shortest Paths Via Quasi-convex Maximization. In: Azar, Y., Erlebach, T. (eds) Algorithms – ESA 2006. ESA 2006. Lecture Notes in Computer Science, vol 4168. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11841036_50
Download citation
DOI: https://doi.org/10.1007/11841036_50
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-38875-3
Online ISBN: 978-3-540-38876-0
eBook Packages: Computer ScienceComputer Science (R0)