Improved Analysis of two Algorithms for Min-Weighted Sum Bin Packing

TL;DR

This paper improves bounds on the approximation ratio of two algorithms for the Min-Weighted Sum Bin Packing problem, a variant relevant to scheduling and logistics, demonstrating the knapsack-batching algorithm's ratio is at most 1.7.

Contribution

The paper provides tighter bounds on the approximation ratios for two algorithms, notably establishing the knapsack-batching algorithm's ratio as at most 1.7.

Findings

Knapsack-batching algorithm has an approximation ratio of at most 1.7.

New lower and upper bounds on algorithm performance are established.

The problem's relevance extends to scheduling and logistics applications.

Abstract

We study the Min-Weighted Sum Bin Packing problem, a variant of the classical Bin Packing problem in which items have a weight, and each item induces a cost equal to its weight multiplied by the index of the bin in which it is packed. This is in fact equivalent to a batch scheduling problem that arises in many fields of applications such as appointment scheduling or warehouse logistics. We give improved lower and upper bounds on the approximation ratio of two simple algorithms for this problem. In particular, we show that the knapsack-batching algorithm, which iteratively solves knapsack problems over the set of remaining items to pack the maximal weight in the current bin, has an approximation ratio of at most 17/10.