License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.MFCS.2020.11
URN: urn:nbn:de:0030-drops-126816
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12681/
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Bannach, Max ; Berndt, Sebastian ; Maack, Marten ; Mnich, Matthias ; Lassota, Alexandra ; Rau, Malin ; Skambath, Malte

Solving Packing Problems with Few Small Items Using Rainbow Matchings

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LIPIcs-MFCS-2020-11.pdf (0.5 MB)

Abstract

An important area of combinatorial optimization is the study of packing and covering problems, such as Bin Packing, Multiple Knapsack, and Bin Covering. Those problems have been studied extensively from the viewpoint of approximation algorithms, but their parameterized complexity has only been investigated barely. For problem instances containing no "small" items, classical matching algorithms yield optimal solutions in polynomial time. In this paper we approach them by their distance from triviality, measuring the problem complexity by the number k of small items.
Our main results are fixed-parameter algorithms for vector versions of Bin Packing, Multiple Knapsack, and Bin Covering parameterized by k. The algorithms are randomized with one-sided error and run in time 4^k⋅ k!⋅ n^{O(1)}. To achieve this, we introduce a colored matching problem to which we reduce all these packing problems. The colored matching problem is natural in itself and we expect it to be useful for other applications. We also present a deterministic fixed-parameter algorithm for Bin Covering with run time O((k!)² ⋅ k ⋅ 2^k ⋅ n log(n)).

BibTeX - Entry

@InProceedings{bannach_et_al:LIPIcs:2020:12681,
  author =	{Max Bannach and Sebastian Berndt and Marten Maack and Matthias Mnich and Alexandra Lassota and Malin Rau and Malte Skambath},
  title =	{{Solving Packing Problems with Few Small Items Using Rainbow Matchings}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{11:1--11:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Javier Esparza and Daniel Kr{\'a}ľ},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12681},
  URN =		{urn:nbn:de:0030-drops-126816},
  doi =		{10.4230/LIPIcs.MFCS.2020.11},
  annote =	{Keywords: Bin Packing, Knapsack, matching, fixed-parameter tractable}
}

Keywords: Bin Packing, Knapsack, matching, fixed-parameter tractable
Collection: 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
Issue Date: 2020
Date of publication: 18.08.2020

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