Abstract
This paper presents an efficient algorithm for minimizing a certain class of submodular functions that arise in analysis of multiclass queueing systems. In particular, the algorithm can be used for testing whether a given multiclass M/M/1 achieves an expected performance by an appropriate control policy. With the aid of the topological sweeping method for line arrangement, our algorithm runs in O(n 2) time, where n is the cardinality of the ground set. This is much faster than direct applications of general submodular function minimization algorithms.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Anagnostou, E., Polimenis, V.G., Guibas, L.J.: Topological sweeping in three dimensions. In: Asano, T., Imai, H., Ibaraki, T., Nishizeki, T. (eds.) SIGAL 1990. LNCS, vol. 450, pp. 310–317. Springer, Heidelberg (1990)
Bertsimas, D., Paschalidis, I.C., Tsitsiklis, J.N.: Optimization of multiclass queueing networks: Polyhedral and nonlinear characterizations of achievable performance. The Annals of Applied Probability 4, 43–75 (1994)
Coffman Jr., E.G., Mitrani, I.: A characterization of waiting time performance realizable by single-server queues. Operations Research 28, 810–821 (1980)
Dacre, M., Glazebrook, K., Niño-Mora, J.: The achievable region approach to the optimal control of stochastic systems. Journal of the Royal Statistical Society, B 61, 747–791 (1999)
Edelsbrunner, H.: Algorithms in Combinatorial Geometry. Springer, Heidelberg (1987)
Edelsbrunner, H., Guibas, L.J.: Topologically sweeping an arrangement. Journal of Computer and Systems Sciences 38, 165–194 (1989)
Edelsbrunner, H., Guibas, L.J.: Topologically sweeping an arrangement — a correction. Journal of Computer and Systems Sciences 42, 249–251 (1991)
Federgruen, A., Groenevelt, H.: Characterization and optimization of achievable performance in general queueing systems. Operations Research 36, 733–741 (1988)
Federgruen, A., Groenevelt, H.: M/G/c queueing systems with multiple customer classes: Characterization and control of achievable performance under nonpreemptive priority rules. Management Science 34, 1121–1138 (1988)
Fujishige, S.: Submodular Function and Optimization. North-Holland, Amsterdam (2005)
Gelenbe, E., Mitrani, I.: Analysis and Synthesis of Computer Systems. Academic Press, London (1980)
Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169–197 (1981)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Heidelberg (1988)
Iwata, S., Fleischer, L., Fujishige, S.: A combinatorial strongly polynomial algorithm for minimizing submodular functions. Journal of the ACM 48, 761–777 (2001)
Kumar, S., Kumar, P.R.: Performance bounds for queueing networks and scheduling policies. IEEE Transactions on Automatic Control 39, 1600–1611 (1994)
McCormick, S.T.: Submodular function minimization. In: Aardal, K., Nemhauser, G., Weismantel, R. (eds.) Handbook on Discrete Optimization, Elsevier, Amsterdam (2005)
Orlin, J.B.: A faster strongly polynomial time algorithm for submodular function minimization. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 240–251. Springer, Heidelberg (2007)
Rafalin, E., Souvaine, D., Streinu, I.: Topological sweep in degenerate cases. In: Mount, D.M., Stein, C. (eds.) ALENEX 2002. LNCS, vol. 2409, pp. 155–165. Springer, Heidelberg (2002), Their implementation is avialable from http://www.cs.tufts.edu/research/geometry/sweep/
Schrijver, A.: A combinatorial algorithm minimizing submodular functions in strongly polynomial time. Journal of Combinatorial Theory, B 80, 346–355 (2000)
Ziegler, G.M.: Lectures on Polytopes. Springer, Heidelberg (1995)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Itoko, T., Iwata, S. (2007). Computational Geometric Approach to Submodular Function Minimization for Multiclass Queueing Systems. In: Fischetti, M., Williamson, D.P. (eds) Integer Programming and Combinatorial Optimization. IPCO 2007. Lecture Notes in Computer Science, vol 4513. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72792-7_21
Download citation
DOI: https://doi.org/10.1007/978-3-540-72792-7_21
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72791-0
Online ISBN: 978-3-540-72792-7
eBook Packages: Computer ScienceComputer Science (R0)