Advances in Operations Research

Volume 2012 (2012), Article ID 543215, 2 pages

doi:10.1155/2012/543215

Erratum

Erratum to “Inapproximability and Polynomial-Time Approximation Algorithm for UET Tasks on Structured Processor Networks”

M. Bouznif¹ and R. Giroudeau²

¹Laboratoire G-SCOP, 46 avenue Félix Viallet, 38031 Grenoble Cedex 1, France

²LIRMM, UMR 5056, 161 rue Ada, 34392 Montpellier Cedex 5, France

Received 1 March 2012; Accepted 27 March 2012

Copyright © 2012 M. Bouznif and R. Giroudeau. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Related Works

This erratum is devoted to give some precisions on the related works in stand of view complexity and approximation given in [1].

1.1. Complexity Results

Table 1 gives the previous complexity results.

Table 1

Topology
Precedence graph
Complexity
Reference

Unbounded chain

-complete
[5]

-complete

Star

-complete
[5]

Cycle/chain

[3]

Star

[6]

1.2. Approximation Results

If the network is a ring, there are two approximation results for Rayward-Smith’s [2] algorithm as follows.

(i)In the general case, the performance ratio is upper bounded by , and there exists an instance for which the performance ratio is equal to [3].(ii) If the number of processors is even, the upper-bound can be improved to , and there exists an instance such that the performance ratio is equal to [4].

The first two corollaries of the paper must be replaced by the two following corollaries.

Previous complexity results on the processors network model.

Corollary 1.1. The problem of deciding whether an instance of ; ; , with has a schedule of length at most three is -complete with .

Proof. See [3].

Moreover, nonapproximability results can be deduced.

Corollary 1.2. No polynomial-time algorithm exists with a performance bound less than unless for the problems ; ; and ; ; , with .

Proof. See [3].

The rest of the paper is devoted to extend the result to the bipartite of depth one and the main complexity result is given in the following theorem and two corollaries.

Theorem 1.3. The problem of deciding whether an instance of , , has a schedule of length at most three is complete.

Corollary 1.4. The problem of deciding whether an instance of ; ; , with has a schedule of length at most three is -complete.

Corollary 1.5. No polynomial-time algorithm exists with a performance bound less than unless for the problems ; ; and ; ; , with .

References

M. Bouznif and R. Giroudeau, “Inapproximability and polynomial-time approximation algorithm for UET tasks on structured processor networks,” Advances in Operations Research, vol. 2011, Article ID 476939, 20 pages, 2011.

V. J. Rayward-Smith, “UET scheduling with unit interprocessor communication delays,” Discrete Applied Mathematics, vol. 18, no. 1, pp. 55–71, 1987.

C. Lahlou, Ordonnancement dans les rseaux de processeurs : complexit et approximation, Ph.D. thesis, Universit Paris VI, 1998.

C. Lahlou, “Scheduling with unit processing and communication times on a ring network: approximation results,” in Proceedings of the Europar, pp. 539–542, Springer, 1996.

C. Picouleau, “UET-UCT schedules on arbitrary networks,” Tech. Rep., LITP, Blaise Pascal, Universit Paris VI, 1994.

R. Giroudeau, J. C. König, and B. Valery, “Scheduling -tasks on a star network: complexity and approximation,” 4OR. A Quarterly Journal of Operations Research, vol. 9, no. 1, pp. 29–48, 2011.