Partitioning to solve Bin Packing Problems

TL;DR

This paper introduces an efficient, easy-to-implement algorithm for the Bin Packing Problem that determines the feasibility of packing items into bins with specific size and item count constraints, ensuring distinct bin arrangements.

Contribution

The paper presents a novel algorithm leveraging number partitioning and multiset concepts to solve a specific bin packing problem with size and count restrictions.

Findings

Algorithm effectively determines packing feasibility under constraints.

Ensures all bins are distinct in the packing arrangement.

Applicable to cases with fixed bin size and item count constraints.

Abstract

The Bin Packing Problem involves efficiently packing items into a limited number of bins without exceeding their capacity. In this paper, we try to answer a specific question in this field. Mathematically the combinatorial optimization problem of this classical Bin Packing can be formulated as follows "Will it be possible to distribute $n$ items with sizes $a_1, \cdots ,a_n$ in total $k$ numbers of bins such that each of the bins consists of exactly $\ell$ number of items whose combined item size is $\mathbb{C}$ provided none of the items left after the allocation? In other words, can we achieve the optimal arrangement $\mathbb{C} =\sum_{i=1}^{n} a_i$ such that $k\ell=n$ also holds under the given size restriction on bins?" We propose an algorithm that determines whether a Bin Packing arrangement is possible while ensuring that each packed bin is distinct. We use tools from standard mathematical concepts such as partitioning of natural number and multiset to derive the algorithm. This algorithm is efficient, easy to implement, and ensures the desired packing in all applicable cases.