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A 3/2-approximation for big two-bar charts packing

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Abstract

We consider a Two-Bar Charts Packing Problem (2-BCPP), in which it is necessary to pack two-bar charts (2-BCs) in a unit-height strip of minimum length. The problem is a generalization of the Bin Packing Problem. Earlier, we proposed an \(O(n^2)\)–time algorithm that constructs the packing of n arbitrary 2-BCs, whose length is at most \(2\cdot OPT+1\), where OPT is the minimum packing length. This paper proposes two new 3/2–approximate algorithms based on sequential matching. One has time complexity \(O(n^4)\) and is applicable when at least one bar of each 2-BC is greater than 1/2. Another has time complexity \(O(n^{3.5})\) and is applicable when, additionally, all BCs are non-increasing or non-decreasing. We prove the estimate’s tightness and conduct a simulation to compare the constructed packings with the optimal solutions or a lower bound of optimum.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    Adil Erzin, Georgii Melidi, Stepan Nazarenko & Roman Plotnikov

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  1. Adil Erzin
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  2. Georgii Melidi
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  3. Stepan Nazarenko
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  4. Roman Plotnikov
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Correspondence to Adil Erzin.

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The research is carried out within the framework of the state contract of the Sobolev Institute of Mathematics (project 0314–2019–0014)

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Erzin, A., Melidi, G., Nazarenko, S. et al. A 3/2-approximation for big two-bar charts packing. J Comb Optim 42, 71–84 (2021). https://doi.org/10.1007/s10878-021-00741-1

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  • DOI: https://doi.org/10.1007/s10878-021-00741-1

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