Two-Bar Charts Packing Problem

Erzin, Adil; Melidi, Gregory; Nazarenko, Stepan; Plotnikov, Roman

doi:10.1007/s11590-020-01657-1

Cited by 11 publications

(21 citation statements)

References 27 publications

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“…The 2-BCPP was first formulated in [11] and then examined in [12,13,14], where the similar problems were described. These problems are the Bin Packing Problem (BPP) [17], the Strip Packing Problem (SPP) [1,8], and the 2-D Vector Packing Problem (2-DVPP) [19].…”

Section: Related Resultsmentioning

confidence: 99%

“…In [12], we proposed an O(n 2 )-time algorithm that builds a packing for n 2-BCs, which length does not exceed 2OP T + 1, where OP T is the minimum packing length for 2-BCPP. When at least one bar's height of each 2-BC is more than 1/2 (such 2-BCs we called "big"), an O(n 3 )-time 3/2-approximation algorithm was proposed.…”

Section: Related Resultsmentioning

confidence: 99%

“…In [12], we formulated 2-BCPP in the form of BLP. However, since in this paper we use the CPLEX package for the BLP to get optimal packings, we replicate its statement below for the reader's convenience.…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…In [12], we had performed an analysis of some approximation algorithms. As a result of the simulation, we concluded that the greedy algorithm GA LO with the preliminary lexicographic ordering of 2-BCs in a non-increasing manner significantly outperforms the other studied approximation algorithms.…”

Section: Algorithmsmentioning

confidence: 99%

“…As a result of the simulation, we concluded that the greedy algorithm GA LO with the preliminary lexicographic ordering of 2-BCs in a non-increasing manner significantly outperforms the other studied approximation algorithms. Therefore, in this article, for comparison with new algorithms, we use only one previously analyzed algorithm GA LO [12].…”

Section: Algorithmsmentioning

confidence: 99%

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Approximation Algorithms for Two-Bar Charts Packing Problem

Erzin¹,

Melidi²,

Nazarenko³

et al. 2021

Preprint

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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.

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