Job Shop Scheduling with Unit Length Tasks: Bounds and Algorithms

We consider the job shop scheduling problem unit−Jm, where each job is processed once on each of m given machines. Every job consists of a permutation of tasks for all machines. The execution of any task on its corresponding machine takes exactly one time unit. The objective is to minimize the overall completion time, called makespan. The contribution of this paper are the following results: (i) For any input instance of unit−Jm with d jobs, the makespan of an optimum schedule is at most m+ o(m), for d = o(m1/2). This improves on the upper bound O(m+ d) given in [LMR99] where O hides a constant equal to two as shown in [S98]. For d = 2 the upper bound is improved to m+⌈ √m ⌉. (ii) There exist input instances of unit−Jm with d = 2 such that the makespan of an optimum schedule is at least m + ⌈ √m ⌉, i.e., our upper bound for d = 2, see result (i), cannot be improved. (iii) We present a randomized on-line approximation algorithm for unit−Jm with the best known approximation ratio for d = o(m1/2). (iv) There is no deterministic on-line algorithm with a competitive ratio better than 4/3 for unit−Jm with two jobs, and for three or more jobs, there is no deterministic on-line algorithm which is better than 1.5 competitive. Compared with the expected competitive ratio of (iii) which tends to 1, this shows that for unit−Jm randomization is very powerful compared with determinism. For two and three jobs, deterministic on-line algorithms with competitive ratios tending to 4/3 and 1.5 respectively are presented. (v) A deterministic approximation algorithm for unit−Jm is described that works in quadratic time for constant d and has an approximation ratio of 1 + 2d/⌊√m ⌋ for d ≤ 2 log2 m.