Point containment in the integer hull of a polyhedron
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Exact and approximate algorithms for high-multiplicity parallel machine scheduling
Journal of Scheduling
Improved approximation algorithms for minimum-space advertisement scheduling
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Mathematics of Operations Research
An OPT+1 algorithm for the cutting stock problem with constant number of object lengths
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Carathéodory bounds for integer cones
Operations Research Letters
Optimal packet-to-slot assignment in mobile telecommunications
Operations Research Letters
An approximate algorithm for a high-multiplicity parallel machine scheduling problem
Operations Research Letters
More on online bin packing with two item sizes
Discrete Optimization
Matching with sizes (or scheduling with processing set restrictions)
Discrete Applied Mathematics
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The following multiprocessor scheduling problem was motivated by scheduling maintenance periods for aircraft. Each maintenance period is a job, and the maintenance facilities are machines. In this context, there are very few different types of maintenances performed, so it is natural to consider the problem with only a small, fixed numberC of different types of jobs. Each job type has a processing time, and each machine is available for the same length of time. A machine can handle at most one job at a time, all jobs are released at time zero, there are no due dates or precedence constraints, and preemption is not allowed. The question is whether it is possible to finish all jobs. We call this problem theMultiprocessor Scheduling Problem with C job lengths (MSPC).Scheduling problems such as MSPC where we can partition the jobs into relatively few types such that all jobs of a certain type are identical are often calledhigh-multiplicity problems. High-multiplicity problems are interesting because their input is very compact: The input to MSPC consists of only 2 C + 2 numbers.For the caseC = 2 we present a polynomial-time algorithm. We show that this algorithm guarantees a schedule that uses the minimum possible number of different one-machine schedules, namely three. Further, we extend this algorithm to the case of machine-dependent deadlines (uniform parallel machines), to a multi-parametric case (that contains the case of unrelated parallel machines), and to some related covering problems. Finally, we give some counterexamples showing why our results do not extend to the caseC 2.