A Tight ($3/2 + \varepsilon$)-Approximation Algorithm for Demand Strip Packing

TL;DR

This work resolves Demand Strip Packing (DSP) with a near-optimal approximation by proving a structural theorem: any instance admits a packing of height at most $\big(\tfrac{3}{2}+\varepsilon\big)\mathrm{OPT}$ that is either $(\varepsilon,\mathrm{OPT})$-neat or $\lambda$-forgiving. The authors introduce Stretching and Squeezing lemmas to repack and insert items while preserving the height bound, enabling two efficient algorithms corresponding to the two structural cases. By running both algorithms and taking the better result, they achieve a polynomial-time $(\tfrac{3}{2}+\varepsilon)$-approximation for any fixed $\varepsilon>0$, matching the NP-hardness barrier up to $\varepsilon$. The approach also sheds light on the related Strip Packing problem and provides techniques (notably repacking and gap-fusing strategies) that could extend to other packing/scheduling problems. Overall, the paper delivers an essentially tight, practically implementable algorithm for DSP with strong structural insight.

Abstract

We consider the Demand Strip Packing problem (DSP), in which we are given a set of jobs, each specified by a processing time and a demand. The task is to schedule all jobs such that they are finished before some deadline $D$ while minimizing the peak demand, i.e., the maximum total demand of tasks executed at any point in time. DSP is closely related to the Strip Packing problem (SP), in which we are given a set of axis-aligned rectangles that must be packed into a strip of fixed width while minimizing the maximum height. DSP and SP are known to be NP-hard to approximate to within a factor below $\frac{3}{2}$. To achieve the essentially best possible approximation guarantee, we prove a structural result. Any instance admits a solution with peak demand at most $\big(\frac32+\varepsilon\big)OPT$ satisfying one of two properties. Either (i) the solution leaves a gap for a job with demand $OPT$ and processing time $\mathcal O(\varepsilon D)$ or (ii) all jobs with demand greater than $\frac{OPT}2$ appear sorted by demand in immediate succession. We then provide two efficient algorithms that find a solution with maximum demand at most $\big(\frac32+\varepsilon\big)OPT$ in the respective case. A central observation, which sets our approach apart from previous ones for DSP, is that the properties (i) and (ii) need not be efficiently decidable: We can simply run both algorithms and use whichever solution is the better one.

Figures (17)

• Figure 1: An optimal packing (left), a $(0,\textsc{Opt})$-neat packing of the same items (middle), and a $\lambda$-forgiving packing (right) of the same items and the extra item $i_\lambda$ (orange).
• Figure 2: One while-loop iteration of the Stretching Algorithm \ref{['alg_stretching']} with the original (left) and the stretched packing (center), and the height bound (right). Removed items (green) are added to $\mathcal{I}_S$. For visualization we also shift the tall item $T$ even though Algorithm \ref{['alg_stretching']} does not pack them.
• Figure 3: An example repacking generated by \ref{['alg_a_pre_fit']} when the tall items have a total width of at least $(1 - \frac{\varepsilon}{5 + 4 \varepsilon}) D$. Here, the blue rectangles depict the same set of wide items.
• Figure 4: An example repacking generated by \ref{['alg_a']} when the tall items have a total width of at least $(1 - \frac{\varepsilon}{5 + 4 \varepsilon}) D$.
• Figure 5: An example repacking generated by \ref{['alg_b_border']} for a segment of width at most $(\frac{1}{2} - \lambda)D$ starting at $0$ that has a width in $[\lambda D,2\lambda D]$ that is not covered by tall items.

Theorems & Definitions (72)

• Theorem 1.1
• Theorem 1.2
• Lemma 1.4: Steinberg Lemma
• Lemma 2.1: Right-Stretching Lemma
• proof
• Corollary 2.2
• Corollary 2.3
• Lemma 2.4: Squeezing Lemma
• proof
• Lemma 4.1