Stochastic Extensible Bin Packing

TL;DR

This paper studies the stochastic extensible bin packing problem, analyzing policy ratios and proposing bounds, with applications to scheduling and stochastic environments, including improved bounds for specific distributions.

Contribution

It introduces tight bounds on the price of non-splittability and fixed assignments, and analyzes a fixed assignment variant of the LEPT rule with near-optimal approximation ratios.

Findings

Price of non-splittability has a tight upper bound of 2.

LEPT variant achieves a tight approximation ratio of approximately 1.368.

Bounds are improved for distributions with bounded Pietra index or stochastic dominance.

Abstract

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 $(1+e^{-1}) \approx 1.368$. 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. We also provide a lower bound on the performance of this policy comparing against an optimal fixed assignment policy. Finally, we obtain improved bounds for the case where the processing times are drawn from a particular family of distributions, with either a bounded Pietra index or when the familly is stochastically dominated at the second order.