Theory of Computing Systems

, 49:834

# Approximation Algorithms for Time-Constrained Scheduling on Line Networks

• Harald Räcke
Article

## Abstract

We consider the problem of time-constrained scheduling of packets in a communication network. Each packet has, in addition to its source and its destination, a release time and a deadline. The goal of an algorithm is to maximize the number of packets that arrive to their destinations by their respective deadlines, given the network constraints.

We consider the line network, and a setting where each node has a buffer of size B packets (where B can be finite or infinite), and each edge has capacity C≥1. To the best of our knowledge this is the first work to study time-constrained scheduling in a setting when buffers can be of limited size. We give approximation algorithms that achieve expected approximation ratio of O(max {log  n−log  B,1}+max {log Σ−log  C,1}), where n is the length of the line, and Σ is the maximum slack a message can have (the slack is the number of time steps a message can be idle and still arrive within its deadline).

A special case of our setting is the setting of buffers of unlimited capacity and edge capacities 1, which has been previously studied by Adler et al. (Theory Comput. Syst. 35(6):599–623, 2002). For this case our results considerably improve upon previous results: We obtain an approximation ratio of $$O(\min\{ \log^{*}n, \log^{*}\Sigma, \log^{*} M\})$$ (where M is the number of messages in the instance), which is a significant improvement upon the results of Adler et al. who obtained a guarantee of O(min {log n,log Σ,log M}).

## Keywords

Approximation algorithms Packet scheduling Time constraints Line networks

## References

1. 1.
Adler, M., Khanna, S., Rajaraman, R., Rosén, A.: Time-constrained scheduling of weighted packets on trees and meshes. Algorithmica 36(2), 123–152 (2003)
2. 2.
Adler, M., Rosenberg, A.L., Sitaraman, R.K., Unger, W.: Scheduling time-constrained communication in linear networks. Theory Comput. Syst. 35(6), 599–623 (2002)
3. 3.
Aiello, W., Kushilevitz, E., Ostrovsky, R., Rosén, A.: Dynamic routing on networks with fixed-size buffers. In: Proceedings of the 14th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 771–780 (2003) Google Scholar
4. 4.
Angelov, S., Khanna, S., Kunal, K.: The network as a storage device: dynamic routing with bounded buffers. Algorithmica 55(1), 71–94 (2009)
5. 5.
Azar, Y., Zachut, R.: Packet routing and information gathering in lines, rings and trees. In: Proceedings of the 13th European Symposium on Algorithms (ESA), pp. 484–495 (2005) Google Scholar
6. 6.
Gordon, E., Rosén, A.: Competitive weighted throughput analysis of greedy protocols on DAGs. ACM Trans. Algorithms 6(3) (2010) Google Scholar
7. 7.
Hoeffding, W.: Probability inequalities for sums of bounded random variables. J. Am. Stat. Assoc. 58(301), 13–30 (1963)
8. 8.
Kleinberg, J.: Approximation algorithms for disjoint paths problems. PhD thesis, Massachusetts Institute of Technology (1996) Google Scholar
9. 9.
Kleinberg, J., Tardos, É.: Approximations for the disjoint paths problem in high-diameter planar network. J. Comput. Syst. Sci. 57(1), 61–73 (1998)
10. 10.
11. 11.
Naor, J.S., Rosén, A., Scalosub, G.: Online time-constrained scheduling in linear and ring networks. J. Discrete Algorithms 8(4), 346–355 (2010)
12. 12.
Panconesi, A., Srinivasan, A.: Randomized distributed edge coloring via an extension of the Chernoff-Hoeffding bounds. SIAM J. Comput. 26(2), 350–368 (1997)
13. 13.
Raghavan, P., Thompson, C.D.: Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica 7(4), 365–374 (1987)
14. 14.
Rexford, J., Hall, J., Shin, K.G.: A router architecture for real-time point-to-point networks. Comput. Archit. News 24(2), 237–246 (1996)