The Price of Fixed Assignments in Stochastic Extensible Bin Packing

• Published in: Approximation and Online Algorithms Vol. 11312; pp. 327 - 347
• Main Authors: Sagnol, Guillaume, Schmidt genannt Waldschmidt, Daniel, Tesch, Alexander
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2018; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Approximation algorithms; extensible bin packing; Stochastic scheduling
• ISBN: 9783030046927; 3030046923
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-030-04693-4_20

We consider the stochastic extensible bin packing problem (SEBP) in which n items of stochastic size are packed into m bins of unit capacity. In contrast to the classical bin packing problem, the number of bins is fixed and they can be extended at extra cost. This problem plays an important role in stochastic environments such as in surgery scheduling: Patients must be assigned to operating rooms beforehand, such that the regular capacity is fully utilized while the amount of overtime is as small as possible. This paper focuses on essential ratios between different classes of policies: First, we consider the price of non-splittability, in which we compare the optimal non-anticipatory policy against the optimal fractional assignment policy. We show that this ratio has a tight upper bound of 2. Moreover, we develop an analysis of a fixed assignment variant of the LEPT rule yielding a tight approximation ratio of $$(1+e^{-1}) \approx 1.368$$ under a reasonable assumption on the distributions of job durations. Furthermore, we prove that the price of fixed assignments, related to the benefit of adaptivity, which describes the loss when restricting to fixed assignment policies, is within the same factor. This shows that in some sense, LEPT is the best fixed assignment policy we can hope for.