Parallel machine scheduling with high multiplicity

Abstract

In high-multiplicity scheduling problems, identical jobs are encoded in the efficient format of describing one of the jobs and the number of identical jobs. Similarly, identical machines are efficiently encoded in the same manner. We investigate parallel-machine, high-multiplicity problems, where there are three possible machine speed structures: identical, proportional, or unrelated. For the objectives of minimizing the sum of job completion times and minimizing the makespan, we consider both nonpreemptive and preemptive problems. For some problems, we develop polynomial time algorithms. For several problems, we demonstrate that the recognition versions can be solved in polynomial time, while the optimization versions require pseudopolynomial time. We also show that changing from standard binary encoding to high-multiplicity encoding does not affect the complexity class of NP-complete problems.

High-multiplicity scheduling on one machine with forbidden start and completion times

“…As a consequence, a polynomial time algorithm under the standard encoding may become exponential under a HM encoding of the instances, which 1 is the case of our algorithms. HM scheduling and more generally HM combinatorial optimization has become an active domain in recent years [3,6,7].…”

A Framework for the Complexity of High-Multiplicity Scheduling Problems

“…We also assume without loss of generality that the jobs are numbered from 1 to n in such a way that jobs 1 to n 1 are of type 1, jobs n 1 + 1 to n 1 + n 2 are of type 2, etc. (Hochbaum and Shamir 1990), (Hochbaum and Shamir 1991), (Hochbaum, Shamir, and Shanthikumar 1992), (Clifford and Posner 2001), an algorithm for SP whose complexity is polynomial in s, L and n is only pseudo-polynomial, but not polynomial in the input size. In order to develop this point more completely, we need to introduce more terminology and notations.…”

“…As in (Papadimitriou and Steiglitz 1982), we now define three distinct scheduling problems associated with F D and f D (see also (Clifford and Posner 2001) …”

“…In spite of these difficulties, many high multiplicity scheduling problems have been proved to be polynomially solvable (see for instance (Brauner, Finke, and Kubiak 2003), (Clifford and Posner 2000), (Clifford and Posner 2001), (Granot and Skorin-Kapov 1993), (Hochbaum and Shamir 1990), (Hochbaum and Shamir 1991), (Hochbaum, Shamir, and Shanthikumar 1992), ( Example 1 (Weighted number of tardy jobs). This is the problem 1|p j = 1| j w j U j .…”

“…(Hochbaum and Shamir 1990), (Hochbaum and Shamir 1991), in particular, have coined the term "high multiplicity" and have underlined the need to discuss the complexity of such problems with special care. A 2 paper by (Clifford and Posner 2001) provides a more detailed framework for this complexity analysis, as well as applications to several specific problems.…”