Abstract
In the problem of Scheduling with Interval Conflicts, there is a ground set of items indexed by integers, and the input is a collection of conflicts, each containing all the items whose index lies within some interval on the real line. Conflicts arrive in an online fashion. A scheduling algorithm must select, from each conflict, at most one survivor item, and the goal is to maximize the number (or weight) of items that survive all the conflicts they are involved in. We present a centralized deterministic online algorithm whose competitive ratio is O(lgσ), where σ is the size of the largest conflict. For the distributed setting, we present another deterministic algorithm whose competitive ratio is \(2\left \lceil {\lg\sigma} \right \rceil \) in the special contiguous case, in which the item indices constitute a contiguous interval of integers. Our upper bounds are complemented by two lower bounds: one that shows that even in the contiguous case, all deterministic algorithms (centralized or distributed) have competitive ratio Ω(lgσ), and that in the non-contiguous case, no deterministic oblivious algorithm (i.e., a distributed algorithm that does not use communication) can have a bounded competitive ratio.
Similar content being viewed by others
Notes
For an efficient implementation (in AC0), e.g. to use in routers, it suffices to extract the smallest bit that is set, using the bit-wise operations (i XOR (i−1)) AND i.
References
Awerbuch, B., Azar, Y., Plotkin, S.A.: Throughput-competitive on-line routing. In: 34th Annual IEEE Symposium on Foundations of Computer Science, pp. 32–40 (1993)
Awerbuch, B., Bartal, Y., Fiat, A., Rosén, A.: Competitive non-preemptive call control. In: Proc. 5th SODA, pp. 312–320 (1994)
Bachmann, U.T., Halldórsson, M.M., Shachnai, H.: Online selection of intervals and t-intervals. In: Proc. 11th SWAT. Lecture Notes in Computer Science, vol. 6139, pp. 383–394 (2010)
Bar-Noy, A., Bar-Yehuda, R., Freund, A., Naor, J., Shieber, B.: A unified approach to approximating resource allocation and scheduling. J. ACM 48(5), 1069–1090 (2001)
Buchbinder, N., Naor, J.: Online primal-dual algorithms for covering and packing. Math. Oper. Res. 34(2), 270–286 (2009)
Emek, Y., Halldórsson, M.M., Mansour, Y., Patt-Shamir, B., Radhakrishnan, J., Rawitz, D.: Online set packing and competitive scheduling of multi-part tasks. In: Proc. 29th PODC, pp. 440–449 (2010)
Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compos. Math. 2, 463–470 (1935)
Golumbic, M.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)
Jaromczyk, J.W., Pezarski, A., Slusarek, M.: An optimal competitive algorithm for the minimal clique covering in circular arc graphs. In: Proc. 19th EWCG (2003)
Karger, D., Stein, C., Wein, J.: Scheduling algorithms. In: Atallah, M.J. (ed.) Algorithms and Theory of Computation Handbook. CRC Press, Boca Raton (1998)
Kesselman, A., Patt-Shamir, B., Scalosub, G.: Competitive buffer management with packet dependencies. In: Proc. 23rd IPDPS, pp. 1–12 (2009)
Lipton, R., Tomkins, A.: Online interval scheduling. In: Proc. 5th SODA, pp. 302–311 (1994)
Sgall, J.: On-line scheduling. In: Online Algorithms. Lecture Notes in Computer Science, vol. 1442, pp. 196–231 (1996)
Author information
Authors and Affiliations
Corresponding author
Additional information
An extended abstract was presented at the 28th International Symposium on Theoretical Aspects of Computer Science (STACS), 2011.
M.M. Halldórsson supported in part by the Icelandic Research Fund (grant 90032021).
B. Patt-Shamir and D. Rawitz research supported in part by the Next Generation Video (NeGeV) Consortium, Israel.
B. Patt-Shamir supported in part by the Israel Science Foundation (grant 1372/09) and by a grant from Israel Ministry of Science and Technology.
Rights and permissions
About this article
Cite this article
Halldórsson, M.M., Patt-Shamir, B. & Rawitz, D. Online Scheduling with Interval Conflicts. Theory Comput Syst 53, 300–317 (2013). https://doi.org/10.1007/s00224-012-9408-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00224-012-9408-1