Abstract
A popular approach in combinatorial optimization is to model problems as integer linear programs. Ideally, the relaxed linear program would have only integer solutions, which happens for instance when the constraint matrix is totally unimodular. Still, sometimes it is possible to build an integer solution with same cost from the fractional solution. Examples are two scheduling problems [4,5] and the single disk prefetching /caching problem [3]. We show that problems such as the three previously mentioned can be separated into two subproblems: (1) finding an optimal feasible set of slots, and (2) assigning the jobs or pages to the slots. It is straigthforward to show that the latter can be solved greedily. We are able to solve the former with a totally unimodular linear program, from which we obtain simple combinatorial algorithms with improved worst case running time.
Supported by CNRS/NSF grant 17171 and ANR Alpage.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms and Applications. Prentice-Hall, Englewood Cliffs (1993)
Albers, S., Büttner, M.: Integrated prefetching and caching in single and parallel disk systems. Information and Computation (198), 24–39 (2005)
Albers, S., Garg, N., Leonardi, S.: Minimizing stall time in single and parallel disk systems. Journal of the ACM (47), 969–986 (2000)
Baptiste, P., Schieber, B.: A note on scheduling tall/small multiprocessor tasks with unit processing time to minimize maximum tardiness. Journal of Scheduling 6(4), 395–404 (2003)
Brucker, P., Kravchenko, S.: Scheduling jobs with equal processing times and time windows on identical parallel machines. Osnabrücker Schriften zur Mathematik H 257, Universität Osnabrück. Fachbereich Mathematik/Informatik. Journal of Scheduling (to appear, 2005)
Cao, P., Felten, E.W., Karlin, A.R., Li, K.: Implementation and performance of integrated application-controlled caching, prefetching and disk scheduling. ACM Transaction of Computer Systems, 188–196 (1995)
Orlin, J.B.: A faster strongly polynomial algorithm for the minimum cost flow problem. Operations Research 41, 338–350 (1993)
Simons, B.: A fast algorithm for single processor scheduling. In: Proceedings IEEE 19th Annual Symposium on Foundations of Computer Science (FOCS 1978), pp. 246–252 (1978)
Simons, B., Warmuth, M.: A fast algorithm for multiprocessor scheduling of unit-length jobs. SIAM Journal on Computing 18(4), 690–710 (1989)
van Leeuwen, J.: HandBook of Theoretical Computer Science. Algorithms and Complexity, vol. A. Elsevier, Amsterdam (1990)
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
Dürr, C., Hurand, M. (2006). Finding Total Unimodularity in Optimization Problems Solved by Linear Programs. 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_30
Download citation
DOI: https://doi.org/10.1007/11841036_30
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)