Efficient Scheduling in Open Shops

Abstract

A new class of polynomially solvable problems within the class of open shop problems is explored. For the problems with m machines and n jobs, an optimal schedule is constructed by an algorithm with the running time polynomial in m and n when there exists a machine, call it i’, whose load M _i’ exceeds the load of any other machine by a given amount of “dominance” M _i’ — M _i’ ≥ (2m - 4)K, i ≠ i’, where K is the maximum processing time of an operation. A weakened dominance condition M _i’ — M _i ≥ (m - 1)K, i ≠ i’, combined with M = max Mi ≥ (5.45m - 7)K, also provides efficient construction of an optimal schedule. In both cases the length of an optimal schedule is M. An approximation O(nm ²) algorithm is also described which, for any function η(m) and any input of the open shop problem satisfying M ≥ η(m)K, yields a schedule with an absolute performance guarantee dependent on the function η(m) and independent of the number of jobs n. In particular, for M > (7m - 6)K, the length of the schedule does not exceed M + (m - 1)K/3, which improves that of any greedy schedule.

This research was partially supported by the Russian Foundation of Fundamental Research (Grant 93–01–00489) and the International Science Foundation (Grant NQC000).