Job Shop Scheduling with Unit Length Tasks: Bounds and Algorithms

Abstract

We consider the job shop scheduling problem unit-J _m, where each job is processed once on each of m given machines. The execution of anyt ask 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 anyi nput instance of unit-J _m with d jobs, the makespan of an optimum schedule is at most m+o(m), for d = o(m ^1/2). For d = o(m ^1/2), this improves on the upper bound O(m+d) given in [LMR99] with a constant equal to two as shown in [S98]. For d = 2 the upper bound is improved to $m + \left\lceil {\sqrt m } \right\rceil$. (ii) There exist input instances of unit-J _m with d = 2 such that the makespan of an optimum schedule is at least $m + \left\lceil {\sqrt m } \right\rceil$, i.e., the result (i) cannot be improved for d = 2. (iii) We present a randomized on-line approximation algorithm for unit-J _m with the best known approximation ratio for d = o(m ^1/2). (iv) A deterministic approximation algorizhm for unit-J _m is described that works in quadratic time for constant d and has an approximation ratio of $1 + 2^d /\left\lfloor {\sqrt m } \right\rfloor$ for d ≤ 2 log₂ m.