Robust Polynomial-Time Approximation Schemes for Parallel Machine Scheduling with Job Arrivals and Departures

Abstract

Scheduling a set of n jobs on m identical parallel machines so as to minimize the makespan or maximize the minimum machine load are two of the most important and fundamental scheduling problems studied in the literature. We consider the general online scenario where jobs are consecutively added to and/or deleted from an instance. The goal is to maintain a near-optimal assignment of the current set of jobs to the m machines. This goal is essentially doomed to failure unless, upon arrival or departure of a job, we allow reassigning some other jobs. Considering that the reassignment of a job induces a cost proportional to its size, the total cost for reassigning jobs must preferably be bounded by a constant r times the total size of added or deleted jobs. The value r is called the reassignment factor of the solution and it is a measure of our willingness to adapt the solution over time. Our main result is that, for any ε > 0, it is possible to achieve (1 + ε)-competitive solutions with constant reassignment factor r(ε). For the minimum makespan problem this is the first improvement on the (2 + ε)-competitive algorithm by Andrews et al. (1999) [Andrews M, Goemans M, Zhang L (1999) Improved bounds for on-line load balancing. Algorithmica 23(4):278–301]. Crucial to our algorithm is a new insight into the structure of robust, almost optimal schedules.

Symmetry Exploitation for Online Machine Covering with Bounded Migration

“…This finally yields a migration factor that only grows polynomially in 1/ε. Finally, we give a lower bound of 17/16 for the best competitive ratio achievable by an algorithm with constant migration, improving the bound on [19].…”

“…Sanders et al [17] give a 2-competitive algorithm with migration factor 1, and this is until now the best competitive ratio known for any algorithm with constant migration factor. On the negative side, Skutella and Verschae [19] show that it is not possible to maintain arbitrarily near optimal solutions using a constant migration factor, giving a lower bound of 20/19 for the best competitive ratio achievable in that case. The lower bound is based on an instance where arriving jobs are very small, not allowing to migrate other jobs.…”

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

“…A general approach in the design of robust online algorithms is to rethink existing algorithmic strategies that work for the corresponding offline problem in a way that the algorithm can adapt to a changing problem instance. The experiences that were made so far in the design of robust algorithms (see [18,4,26]) are to design the algorithm in a way such that the generated solutions fulfill very tight structural properties. Such solutions can then be adapted more easily as new items arrive.…”

“…Such difficulties also arise in related optimization problems, see e.g. [26,4]. For the case of flat items (small height) we overcome these difficulties by the packing structure shown in Figure 1a: Flat items are separated from big items in the containers and are sorted by width such that the least wide item is at the top.…”

“…We say that an algorithm has an amortized migration factor of µ if for every time step t the total migration (i. e. the total area of repacked items) up to time t is bounded by µ t j=1 SIZE(i j ), where SIZE(i j ) is the area of item i j arrived at time j. Adapted to scheduling problems this corresponds with the reassignment cost model introduced by Skutella and Verschae in [26]. We adapt several offline and online techniques and combine them with our novel approaches to obtain the following main result: Theorem 1.2.…”