Variable length sequencing with two lengths

Certain tasks, like accessing pages on the World Wide Web, require duration that varies over time. This poses the following Variable Length Sequencing Problem, or VLSP-L. Let [i, j) denote {i, i+1, ..., j- 1}. The problem is given by a set of jobs J and the time-dependent length function λ : J ×[0, n) → L. A sequencing function σ :J →[0, n) assigns to each job j a time interval Τσ(j) when this job is executed; if σ(j) = t then Τσ(j) = [t, t+λ(j, t)). The sequencing is valid if these time intervals are disjoint. Our objective is to minimize the makespan, i. e. the maximum ending of an assign time interval. Recently it was shown VLSP-[0, n) is NP-hard and that VLSP-{1, 2} can be solved efficiently. For a more general case of VLSP-{1, k} an 2 - 1/k approximation was shown. This paper shows that for k ≥ 3 VLSP-{1, k} is MAX-SNP hard, and that we can approximate it with ratio 2 - 4/(k + 3).