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

TL;DR

This paper presents a near-optimal approximation algorithm for Demand Strip Packing, achieving a $(3/2 + ext{epsilon})$ factor by leveraging a novel structural property of solutions.

Contribution

The authors introduce a structural result that enables a $(3/2 + ext{epsilon})$-approximation algorithm for DSP, improving upon previous bounds and providing two efficient algorithms for different solution cases.

Findings

Achieves a $(3/2 + ext{epsilon})$-approximation for DSP.

Introduces a structural property of solutions for DSP.

Provides two algorithms that find near-optimal solutions based on the structural property.

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.