Green Bin Packing

TL;DR

GBP extends online bin packing by introducing a two-tier cost: opening a new bin costs 1, while using black space beyond a green threshold incurs a linear cost at rate $\beta$. The paper shows offline GBP is NP-hard yet admits an APTAS, and develops threshold-based online algorithms that adapt to the regimes $\beta G\le 1$ and $\beta G>1$, with both worst-case and empirical analyses. The main contributions are explicit competitive-ratio bounds for classical and thresholded algorithms, a refined weight-per-cost methodology, and extensive experiments across Weibull, Uniform, and GI distributions, illustrating practical guidance for data-center load balancing and overcommitment scenarios. The results highlight the two-tier cost trade-off shaping algorithm design, with WorstFit and thresholded variants often outperforming classical baselines in GBP’s online setting. The work thus advances sustainable resource allocation planning and motivates future work on adaptive thresholds, learning-augmented GBP, and nonlinear cost models.

Abstract

The online bin packing problem and its variants are regularly used to model server allocation problems. Modern concerns surrounding sustainability and overcommitment in cloud computing motivate bin packing models that capture costs associated with highly utilized servers. In this work, we introduce the green bin packing problem, an online variant with a linear cost $β$ for filling above a fixed level $G$. For a given instance, the goal is to minimize the sum of the number of opened bins and the linear cost. We show that when $βG \le 1$, classical online bin packing algorithms such as FirstFit or Harmonic perform well, and can achieve competitive ratios lower than in the classic setting. However, when $βG > 1$, new algorithmic solutions can improve both worst-case and typical performance. We introduce variants of classic online bin packing algorithms and establish theoretical bounds, as well as test their empirical performance.

Figures (12)

• Figure 1: Competitive ratio bounds for $\beta G \le 1$. Top row displays values of $G \le 1/2$, bottom row displays $G >1/2$
• Figure 2: General Online Lower Bound vs $\texttt{AlmostAnyFit}$/$\texttt{Harmonic}$ Lower Bound for $\beta G \le 1$ (a and b). The general online bound is shown with dotted lines, while $\texttt{AlmostAnyFit}$/$\texttt{Harmonic}$ is shown with solid lines. And (c) gives a comparison of the General Online Lower Bound to the upper bound of $\texttt{AlmostAnyFit}_\tau$ for $\beta G > 1$ and $G\le 1/2$.
• Figure 3: Experiments on Weibull distribution with fixed $\beta$ for $\beta G \le 1$
• Figure 4: Experiments on Weibull distribution with $G=1/2$. (a): All $\tau=0$ (b): All $\tau$ set to empirically best $\tau$ (c): All $\tau$ set to theoretical best $\tau$ for given $\beta G$
• Figure 5: Experiments on Weibull distribution with fixed $G$ using empirically determined $\tau$.

Theorems & Definitions (71)

• proposition 1
• proposition 2
• proof
• lemma 1
• proof
• theorem 1: Lower Bound of Offline GBP
• proof
• theorem 2
• theorem 3
• theorem 4