A $4/3\cdot OPT+2/3$ approximation for big two-bar charts packing problem

TL;DR

This paper introduces a new approximation algorithm with a ratio of 4/3·OPT+2/3 for packing two-bar charts into minimal bins, extending previous results and addressing NP-hardness in this problem.

Contribution

It presents the first approximation algorithm with a ratio of 4/3·OPT+2/3 for the two-bar charts packing problem, improving upon prior approaches.

Findings

The problem remains strongly NP-hard even with bars taller than 1/2.

A greedy algorithm achieves a packing within OPT+1 for certain cases.

The main algorithm guarantees a packing within 4/3·OPT+2/3 for the general NP-hard case.

Abstract

Two-Bar Charts Packing Problem is to pack $n$ two-bar charts (2-BCs) in a minimal number of unit-capacity bins. This problem generalizes the strongly NP-hard Bin Packing Problem. We prove that the problem remains strongly NP-hard even if each 2-BC has at least one bar higher than 1/2. Next we consider the case when the first (or second) bar of each 2-BC is higher than 1/2 and show that the $O(n^2)$-time greedy algorithm with preliminary lexicographic ordering of 2-BCs constructs a packing of length at most $OPT+1$, where $OPT$ is optimum. Eventually, this result allowed us to present an $O(n^{2.5})$-time algorithm that constructs a packing of length at most $4/3\cdot OPT+2/3$ for the NP-hard case when each 2-BC has at least one bar higher than 1/2.