Approximation Algorithms for Two-Bar Charts Packing Problem

TL;DR

This paper evaluates approximation algorithms for the Two-Bar Charts Packing Problem by comparing their solutions to optimal solutions obtained through Boolean Linear Programming, revealing inaccuracies in previous guarantees.

Contribution

It provides a comprehensive numerical analysis of approximation algorithms for 2-BCPP, highlighting their performance and limitations compared to optimal solutions.

Findings

Approximation algorithms' guarantees are often inaccurate.

Optimal solutions were computed using Boolean Linear Programming.

The study offers a benchmark for future algorithm assessments.

Abstract

In the Two-Bar Charts Packing Problem (2-BCPP), it is required to pack the bar charts (BCs) consisting of two bars into the horizontal unit-height strip of minimal length. The bars may move vertically within the strip, but it is forbidden to change the order and separate the chart's bars. Recently, for this new problem, which is a generalization of the Bin Packing Problem (BPP), Strip Packing Problem (SPP), and 2-Dimensional Vector Packing Problem (2-DVPP), several approximation algorithms with guaranteed estimates were proposed. However, after a preliminary analysis of the solutions constructed by approximation algorithms, we discerned that the guaranteed estimates are inaccurate. This fact inspired us to conduct a numerical experiment in which the approximate solutions are compared to each other and with the optimal ones. To construct the optimal solutions or lower bounds for optimum, we use the Boolean Linear Programming (BLP) formulation of 2-BCPP proposed earlier and apply the CPLEX package. We also use a database of instances for BPP with known optimal solutions to construct the instances for the 2-BCPP with known minimal packing length. The results of the simulation make up the main content of this paper.