Approximation algorithms for the square min-sum bin packing problem

TL;DR

This paper introduces approximation algorithms for the square min-sum bin packing problem, minimizing total packing cost, and demonstrates the limitations of classic heuristics while providing a PTAS and a specific approximation ratio.

Contribution

The paper presents a $rac{53}{22}$-approximation algorithm and a PTAS for SMSBPP, improving solution approaches for this problem.

Findings

Classic bin packing heuristics can perform arbitrarily poorly on SMSBPP.

A new $rac{53}{22}$-approximation algorithm is proposed.

A Polynomial-Time Approximation Scheme (PTAS) is developed for SMSBPP.

Abstract

In this work, we study the square min-sum bin packing problem (SMSBPP), where a list of square items has to be packed into indexed square bins of dimensions $1 \times 1$ with no overlap between the areas of the items. The bins are indexed and the cost of packing each item is equal to the index of the bin in which it is placed in. The objective is to minimize the total cost of packing all items, which is equivalent to minimizing the average cost of items. The problem has applications in minimizing the average time of logistic operations such as cutting stock and delivery of products. We prove that classic algorithms for two-dimensional bin packing that order items in non-increasing order of size, such as Next Fit Decreasing Height or Any Fit Decreasing Height heuristics, can have an arbitrarily bad performance for SMSBPP. We, then, present a $\frac{53}{22}$-approximation and a PTAS for the problem.