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

TL;DR

This paper investigates the computational complexity and approximation algorithms for a two-dimensional demand bin packing problem, where tasks are scheduled over time with capacity constraints, providing hardness results and effective algorithms.

Contribution

It establishes NP-hardness of approximation within a factor of 2 for certain variants and offers the best possible algorithms, including a 3-approximation for the general case.

Findings

NP-hardness of approximation within factor 2 for specific variants

Optimal approximation algorithms for variants with short height and square tasks

A simple 3-approximation algorithm for the general 2D demand bin packing problem

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.