Max-Min and 1-Bounded Space Algorithms for the Bin Packing Problem

TL;DR

This paper analyzes the Max-Min and 1-bounded space algorithms for the 1-dimensional bin packing problem, proving an approximation ratio of 1.5 for the MM algorithm and exploring theoretical bounds for related algorithm classes.

Contribution

It establishes the asymptotic approximation ratio of the MM algorithm as at most 1.5 and provides bounds for the intersection of max-min and 1-bounded space algorithm classes.

Findings

Approximation ratio of MM algorithm is at most 1.5.

Lower bound of 1.25 for the intersection of algorithm classes.

Extended analysis to cardinality constrained bin packing.

Abstract

In the (1-dimensional) bin packing problem, we are asked to pack all the given items into bins, each of capacity one, so that the number of non-empty bins is minimized. Zhu~[Chaos, Solitons \& Fractals 2016] proposed an approximation algorithm $MM$ that sorts the item sequence in a non-increasing order by size at the beginning, and then repeatedly packs, into the current single open bin, first as many of the largest items in the remaining sequence as possible and then as many of the smallest items in the remaining sequence as possible. In this paper we prove that the asymptotic approximation ratio of $MM$ is at most 1.5. Next, focusing on the fact that $MM$ is at the intersection of two algorithm classes, max-min algorithms and 1-bounded space algorithms, we comprehensively analyze the theoretical performance bounds of each subclass derived from the two classes. Our results include a lower bound of 1.25 for the intersection of the two classes. Furthermore, we extend the theoretical analysis over algorithm classes to the cardinality constrained bin packing problem.