The Power of Migrations in Dynamic Bin Packing

TL;DR

The paper investigates online dynamic bin packing with migrations, focusing on minimizing total active time and the impact of limited migrations. It establishes a sublinear-versus-linear dichotomy: sublinear migrations offer no substantial advantage over zero migrations, while a linear fraction of migrations yields a tight $\approx 1/\alpha$ trade-off, with extensions to size-cost migrations. It then introduces a delay-cost migration model DynBinPack-Delay, achieving an $O(\min(\sqrt{C}, \mu))$-approximation and a matching lower bound, where $C$ is the per-migration delay and $\mu$ is the duration ratio. Together, these results map the power of migrations across regimes and provide practical, delay-aware strategies for dynamic resource packing in systems such as cloud/job scheduling.

Abstract

In the Dynamic Bin Packing problem, $n$ items arrive and depart the system in an online manner, and the goal is to maintain a good packing throughout. We consider the objective of minimizing the total active time, i.e., the sum of the number of open bins over all times. An important tool for maintaining an efficient packing in many applications is the use of migrations; e.g., transferring computing jobs across different machines. However, there are large gaps in our understanding of the approximability of dynamic bin packing with migrations. Prior work has covered the power of no migrations and $> n$ migrations, but we ask the question: What is the power of limited ($\leq n$) migrations? Our first result is a dichotomy between no migrations and linear migrations: Using a sublinear number of migrations is asymptotically equivalent to doing zero migrations, where the competitive ratio grows with $μ$, the ratio of the largest to smallest item duration. On the other hand, we prove that for every $α\in (0,1]$, there is an algorithm that does $\approx αn$ migrations and achieves competitive ratio $\approx 1/α$ (in particular, independent of $μ$); we also show that this tradeoff is essentially best possible. This fills in the gap between zero migrations and $> n$ migrations in Dynamic Bin Packing. Finally, in light of the above impossibility results, we introduce a new model that more directly captures the impact of migrations. Instead of limiting the number of migrations, each migration adds a delay of $C$ time units to the item's duration; this commonly appears in settings where a blackout or set-up time is required before the item can restart its execution in the new bin. In this new model, we prove a $O(\min (\sqrt{C}, μ))$-approximation, and an almost matching lower bound.

Figures (2)

• Figure 1: Illustration of an item with arrival time $a_0$ and duration $d_0$ with 0 and 2 migrations. Each migration leads to a delay of length $C$ in its total duration.
• Figure 2: $k^2$ items of size $\frac{1}{k}$ arrive at time 0. The algorithm puts them in $k$ bins without knowing their duration. The adversary picks one item on each bin to be long-lived (duration $\mu$) and all others short-lived (duration 1). The cost of the algorithm is $k \cdot \mu$. An optimal allocation puts all long-lived items on one bin and has cost $\mu + (k-1)$. The competitive ratio $\frac{k \cdot \mu}{\mu + (k-1)}$ goes to $\mu$ as $k\rightarrow \infty$.

Theorems & Definitions (41)

• Theorem 1: \ref{['thm:sublinear']} (Informal)
• Theorem 2
• Theorem 3
• Theorem 4: Informal
• Theorem 5
• Theorem 6
• Corollary 1
• Lemma 1
• proof
• Theorem 7