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Algorithmica

pp 1–15 | Cite as

Parameterized Algorithms and Kernels for Rainbow Matching

  • Sushmita Gupta
  • Sanjukta Roy
  • Saket Saurabh
  • Meirav Zehavi
Article

Abstract

In this paper, we study the NP-complete colorful variant of the classical Matching problem, namely, the Rainbow Matching problem. Given an edge-colored graph G and a positive integer k, this problem asks whether there exists a matching of size at least k such that all the edges in the matching have distinct colors. We first develop a deterministic algorithm that solves Rainbow Matching on paths in time \(\mathcal{O}^\star \left( \left( \frac{1+\sqrt{5}}{2}\right) ^k\right) \) and polynomial space. This algorithm is based on a curious combination of the method of bounded search trees and a “divide-and-conquer-like” approach, where the branching process is guided by the maintenance of an auxiliary bipartite graph where one side captures “divided-and-conquered” pieces of the path. Our second result is a randomized algorithm that solves Rainbow Matching on general graphs in time \(\mathcal {O} ^\star (2^k)\) and polynomial-space. Here, we show how a result by Björklund et al. (J Comput Syst Sci 87:119–139, 2017) can be invoked as a black box, wrapped by a probability-based analysis tailored to our problem. We also complement our two main results by designing kernels for Rainbow Matching on general and bounded-degree graphs.

Keywords

Rainbow matching Parameterized algorithm Bounded search trees Divide-and-conquer 3-Set packing 3-Dimensional matching 

Notes

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of BergenBergenNorway
  2. 2.The Institute of Mathematical SciencesHBNIChennaiIndia
  3. 3.Ben-Gurion UniversityBeershebaIsrael

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