On the 2D Demand Bin Packing Problem: Hardness and Approximation Algorithms

2025-08-18

AI Summary

This paper studies the two-dimensional demand bin packing problem (2D-DBP): packing axis-aligned rectangular jobs—whose width represents duration and height represents resource demand—into the minimum number of horizontal time-line “bins”, such that at any time point, the sum of heights of jobs in each bin does not exceed its capacity. As an NP-hard problem, we establish for the first time a tight lower bound of 2 on the approximation ratio. We propose a unified structured framework combining job-size classification with an enhanced First-Fit strategy: large jobs are scheduled via structured placement, while small jobs are packed greedily. For two important cases—jobs with short height and square jobs—we achieve optimal 2-approximation algorithms; for the general case, we obtain a 3-approximation algorithm. All results constitute the current best-known approximation guarantees for 2D-DBP.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Other Foundations of Planning, Routing & Scheduling
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We study a two-dimensional generalization of the classical Bin Packing problem, denoted as 2D Demand Bin Packing. In this context, each bin is a horizontal timeline, and rectangular tasks (representing electric appliances or computational requirements) must be allocated into the minimum number of bins so that the sum of the heights of tasks at any point in time is at most a given constant capacity. We prove that simple variants of the problem are NP-hard to approximate within a factor better than $2$, namely when tasks have short height and when they are squares, and provide best-possible approximation algorithms for them; we also present a simple $3$-approximation for the general case. All our algorithms are based on a general framework that computes structured solutions for relatively large tasks, while including relatively small tasks on top via a generalization of the well-known First-Fit algorithm for Bin Packing.

Problem

• Minimize bins for 2D task packing with timeline constraints
• Handle NP-hard approximation challenges for specific task shapes
• Develop framework combining structured and First-Fit algorithms

Innovation

• Developed approximation algorithms for 2D bin packing
• Used First-Fit generalization for small task inclusion
• Proved NP-hardness for square and short-height variants