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Publisher DOI: 10.4230/LIPIcs.MFCS.2020.11
arXiv ID: 2007.02660
Title: Solving packing problems with few small items using rainbow matchings
Language: English
Authors: Bannach, Max 
Berndt, Sebastian 
Maack, Marten 
Mnich, Matthias  
Lassota, Alexandra 
Rau, Malin 
Skambath, Malte 
Keywords: Bin Packing; Knapsack; matching; fixed-parameter tractable
Issue Date: 2020
Source: International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
Abstract (english): 
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 4k · k! · nO(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 Packing with run time O((k!)2 · k · 2k · n log(n)).
Conference: 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020) 
DOI: 10.15480/882.2882
ISBN: 978-3-95977-159-7
Institute: Algorithmen und Komplexität E-11 
Document Type: Chapter/Article (Proceedings)
Project: Kernelisierung für große Datenmengen 
Multivariate Algorithmen für Scheduling mit hoher Multiplizität 
License: CC BY 3.0 (Attribution) CC BY 3.0 (Attribution)
Part of Series: Leibniz international proceedings in informatics (LIPIcs) 
Volume number: 170
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