Abstract
We consider the problem of scheduling jobs that are given as groups of non-intersecting intervals on the real line. Each job j is associated with a t-interval, which consists of up to t segments, for some t ≥ 1, a demand, d j ∈ [0,1], and a weight, w(j). A schedule is a collection of jobs, such that, for every \(s \in {\mathbb R}\), the total demand of the jobs in the schedule whose t-interval contains s does not exceed 1. Our goal is to find a schedule that maximizes the total weight of scheduled jobs.
We present a 6t-approximation algorithm that uses a novel extension of the primal-dual schema called fractional primal-dual. The first step in a fractional primal-dual r-approximation algorithm is to compute an optimal solution, x *, of an LP relaxation of the problem. Next, the algorithm produces an integral primal solution x, and a new LP, denoted by P′, that has the same objective function as the original problem, but contains inequalities that may not be valid with respect to the original problem. Moreover, x * is a feasible solution of P′. The algorithm also computes a solution y to the dual of P′. x is r-approximate, since its weight is bounded by the value of y divided by r.
We present a fractional local ratio interpretation of our 6t-approximation algorithm. We also discuss the connection between fractional primal-dual and the fractional local ratio technique. Specifically, we show that the former is the primal-dual manifestation of the latter.
Keywords
- Interval Graph
- Local Ratio
- Complementary Slackness Condition
- Optimal Fractional Solution
- Linear Resource
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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
Bar-Yehuda, R., Halldórsson, M.M., Naor, J., Shachnai, H., Shapira, I.: Scheduling split intervals. In: 13th Annual Symposium on Discrete Algorithms, pp. 732–741 (2002)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)
Halldórsson, M.M., Rajagopalan, S., Shachnai, H., Tomkins, A.: Shceduling multiple resources (1999) (Manuscript)
Rotem, D.: Analysis of disk arm movement for large sequential reads. In: 11th ACM Symposium on Principles of Database Systems, pp. 47–54 (1992)
Bafna, V., Narayanan, B.O., Ravi, R.: Nonoverlapping local alignments (weighted independent sets of axis parallel rectangles). Disc. Appl. Math. 71, 41–53 (1996)
Berman, P., Fujito, T.: Approximating independent sets in degree 3 graphs. In: 4th Workshop on Algorithms and Data Structures. LNCS, vol. 995, pp. 449–460. Springer, Heidelberg (1995)
Halldórsson, M.M., Yoshihara, K.: Greedy approximations of independent sets in low degree graphs. In: Staples, J., Katoh, N., Eades, P., Moffat, A. (eds.) ISAAC 1995. LNCS, vol. 1004, pp. 152–161. Springer, Heidelberg (1995)
Hazan, E., Safra, S., Schwartz, O.: On the hardness of approximating k-dimensional matching. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 83–97. Springer, Heidelberg (2003)
Hurkens, C.A.J., Schrijver, A.: On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems. SIAM Journal on Discrete Mathematics 2, 68–72 (1989)
Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation and complexity classes. Journal of Computer and System Sciences 43, 425–440 (1991)
West, D.B., Shmoys, D.B.: Recognizing graphs with fixed interval number is NP-complete. Discrete Applied Mathematics 8, 295–305 (1984)
Gyárfás, A., West, D.B.: Multitrack interval graphs. In: 26th SE Intl. Conf. Graph Th. Comb. Comput. Congr. Numer., vol. 109, pp. 109–116 (1995)
Bar-Yehuda, R., Even, S.: A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics 25, 27–46 (1985)
Bar-Noy, A., Bar-Yehuda, R., Freund, A., Naor, J., Shieber, B.: A unified approach to approximating resource allocation and schedualing. J. ACM 48, 1069–1090 (2001)
Bar-Yehuda, R., Rawitz, D.: On the equivalence between the primal-dual schema and the local ratio technique. SIAM J. on Disc. Math. (2005) (to appear)
Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM J. on Comp. 24, 296–317 (1995)
Goemans, M.X., Williamson, D.P.: The primal-dual method for approximation algorithms and its application to network design problems. In: Hochbaum, D.S. (ed.) Approximation Algorithms for NP-Hard Problem. PWS Publishing Company (1997)
Williamson, D.P.: The primal dual method for approximation algorithms. Mathematical Programming 91, 447–478 (2002)
Bertsimas, D., Teo, C.: From valid inequalities to heuristics: A unified view of primal-dual approximation algorithms in covering problems. Oper. Res. 46, 503–514 (1998)
Bafna, V., Berman, P., Fujito, T.: A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM J. on Disc. Math. 12, 289–297 (1999)
Becker, A., Geiger, D.: Optimization of Pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artificial Intelligence 83, 167–188 (1996)
Chudak, F.A., Goemans, M.X., Hochbaum, D.S., Williamson, D.P.: A primal-dual interpretation of recent 2-approximation algorithms for the feedback vertex set problem in undirected graphs. Oper. Res. Lett. 22, 111–118 (1998)
Lewin-Eytan, L., Naor, J., Orda, A.: Admission control in networks with advance reservations. Algorithmica 40, 293–403 (2004)
Bar-Yehuda, R.: One for the price of two: A unified approach for approximating covering problems. Algorithmica 27, 131–144 (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bar-Yehuda, R., Rawitz, D. (2005). Using Fractional Primal-Dual to Schedule Split Intervals with Demands. In: Brodal, G.S., Leonardi, S. (eds) Algorithms – ESA 2005. ESA 2005. Lecture Notes in Computer Science, vol 3669. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11561071_63
Download citation
DOI: https://doi.org/10.1007/11561071_63
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-29118-3
Online ISBN: 978-3-540-31951-1
eBook Packages: Computer ScienceComputer Science (R0)