On polynomial-time approximation algorithms for the variable length scheduling problem

This paper may be viewed as a corrigendum as well as an extension of the paper by (Czumaj et al., Theoret. Comput. Sci. 262 (1–2), (2001) 569–582) where they deal with the variable length scheduling problem (VLSP) with parameters k1; k2, denoted VLSP(k1; k2). In the current paper, we >rst discuss an error in the analysis of one of the approximation algorithms described in (Czumaj et al., Theoret. Comput. Sci. 262 (1–2), (2001) 569–582), where an approximation algorithm for VLSP(k1; k2), k1 ¡k2, was presented and it was claimed that the algorithm achieves the approximation ratio of 1 + (k1(k2 − k1))=k2. In this paper we give a problem instance for which the same algorithm obtains the approximation ratio ≈ k2 k1 . We then present two simple approximation algorithms, one for the case k1 = 1 with an approximation ratio of 2, and one for the case k1 ¿ 1 with an approximation ratio of 2+ (k2=2k1). This corrects the result claimed in (Czumaj et al., Theoret. Comput. Sci. 262 (1–2), (2001) 569–582). c © 2003 Elsevier Science B.V. All rights reserved. ∗ Corresponding author. E-mail addresses: [email protected] (A. Czumaj), [email protected] (L. G0 asieniec), [email protected] (D.R. Gaur), [email protected] (R. Krishnamurti), [email protected] (W. Rytter), [email protected] (M. Zito). 0304-3975/03/$ see front matter c © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0304-3975(03)00141-5 490 A. Czumaj et al. / Theoretical Computer Science 302 (2003) 489–495