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arXiv · 2212.00944

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

Adil Erzin·Alexander Kononov·Georgii Melidi·Stepan Nazarenko
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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.

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Adil Erzin, Alexander Kononov, Georgii Melidi, Stepan Nazarenko. 2022-12-02. A $4/3\cdot OPT+2/3$ approximation for big two-bar charts packing problem. https://arxiv.org/abs/2212.00944

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