Two-Bar Charts Packing Problem

TL;DR

This paper introduces a polynomial-time approximation algorithm for the Two-Bar Charts Packing Problem, a generalization of bin packing, providing the first guaranteed estimate and demonstrating high efficiency through numerical experiments.

Contribution

It presents the first guaranteed approximation algorithm for 2-BCPP with a proven bound and validates its effectiveness via computational experiments.

Findings

Algorithm achieves a packing length of at most 2*OPT+1

Numerical experiments show high efficiency compared to optimal solutions

First known guaranteed estimate for 2-BCPP

Abstract

We consider a Bar Charts Packing Problem (BCPP), in which it is necessary to pack bar charts (BCs) in a strip of minimum length. The problem is, on the one hand, a generalization of the Bin Packing Problem (BPP), and, on the other hand, a particular case of the Project Scheduling Problem with multidisciplinary jobs and one limited non-accumulative resource. Earlier, we proposed a polynomial algorithm that constructs the optimal package for a given order of non-increasing BCs. This result generalizes a similar result for BPP. For Two-Bar Charts Packing Problem (2-BCPP), when each BC consists of two bars, the algorithm we have proposed constructs a package in polynomial time, the length of which does not exceed $2\ OPT+1$, where $OPT$ is the minimum possible length of the packing. As far as we know, this is the first guaranteed estimate for 2-BCPP. We also conducted a numerical experiment in which we compared the solutions built by our approximate algorithms with the optimal solutions built by the CPLEX package. The experimental results confirmed the high efficiency of the developed algorithms.