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Parameterized Algorithms and Kernels for Rainbow Matching

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Sushmita Gupta, Sanjukta Roy, Saket Saurabh, and Meirav Zehavi. Parameterized Algorithms and Kernels for Rainbow Matching. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 71:1-71:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.MFCS.2017.71

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 O*(((1+\sqrt{5})/2)^k) 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 O*(2^k) and polynomial space. Here, we show how a result by Björklund et al. [JCSS, 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

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