Michaël Gabay

An efficient algorithm for semi-online multiprocessor scheduling with given total processing time

We consider a semi-online multiprocessor scheduling problem with a given a set of identical machines and a sequence of jobs, the sum of whose processing times is known in advance. The jobs are to be assigned online to one of the machines and the objective is to minimize the makespan. The best known algorithm for this problem achieves a competitive ratio 1.6 (Cheng et al. in Theor Comput Sci 337:134–146, 2005). The best known lower bound is approximately 1.585 (Albers and Hellwig in Theor Comput Sci 443:1–9, 2012) if the number of machines tends to infinity. We present an elementary algorithm with competitive ratio equal to this lower bound. Thus, the algorithm is best possible if the number of machines tends to infinity.

Vector bin packing with heterogeneous bins: application to the machine reassignment problem

In this paper, we introduce a generalization of the vector bin packing problem, where the bins have variable sizes. This generalization can be used to model virtual machine placement problems and in particular to build feasible solutions for the machine reassignment problem. We propose several families of greedy heuristics for this problem and show that they are flexible and can be adapted to handle additional constraints. We present structural properties of the machine reassignment problem, that allow us to decompose it into smaller subproblems and adapt our heuristics to them. We evaluate our heuristics on academic benchmarks of the vector bin packing problem, a randomly generated vector bin packing problem with heterogeneous bins benchmark as well as Google’s realistic instances of the machine reassignment problem.

Online bin stretching with bunch techniques

We are given a sequence of items that can be packed into m unit size bins and the goal is to assign these items online to m bins while minimizing the stretching factor. Bins have infinite capacities and the stretching factor is the size of the largest bin. We present an algorithm with stretching factor 26 / 17 ? 1.5294 improving the best known algorithm by Kellerer and Kotov (2013) 1 with a stretching factor 11 / 7 ? 1.5714 . Our algorithm has 2 stages and uses bunch techniques: we aggregate bins into batches sharing a common purpose.

Improved lower bounds for the online bin stretching problem

We use game theory techniques to automatically compute improved lower bounds on the competitive ratio for the bin stretching problem. Using these techniques, we improve the best lower bound for this problem to 19/14. We explain the technique and show that it can be generalized to compute lower bounds for any online or semi-online packing or scheduling problem.

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

We are interested in a single machine scheduling problem where jobs can neither start nor end on some specified instants, and the aim is to minimize the makespan. This problem models the situation where an additional resource, subject to unavailability constraints, is required to start and to finish a job. We consider in this paper the high-multiplicity version of the problem, when the input is given using a compact encoding. We present a polynomial time algorithm for large diversity instances (when the number of different processing times is greater than the number of forbidden instants). We also show that this problem is fixed-parameter tractable when the number of forbidden instants is fixed, regardless of jobs characteristics.

High Multiplicity Scheduling with Switching Costs for Few Products

We study several variants of the single machine capacitated lot sizing problem with sequence-dependent setup costs and product-dependent inventory costs. Here we are given one machine and n≥1n≥1 types of products that need to be scheduled. Each product is associated with a constant demand rate didi, production rate pipi and inventory costs per unit hihi. When the machine switches from producing product i to product j, setup costs si,jsi,j are incurred. The goal is to minimize the total costs subject to the condition that all demands are satisfied and no backlogs are allowed. In this work, we show that by considering the high multiplicity setting and switching costs, even trivial cases of the corresponding “normal” counterparts become non-trivial in terms of size and complexity. We present solutions for one and two products.