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Parameterized Algorithms and Kernels for Rainbow Matching

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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.

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Notes

  1. Specifically, if the algorithm determines that an input instance is a yes-instance, then this answer is necessarily correct.

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Authors and Affiliations

  1. University of Bergen, Bergen, Norway

    Sushmita Gupta & Saket Saurabh

  2. The Institute of Mathematical Sciences, HBNI, Chennai, India

    Sanjukta Roy & Saket Saurabh

  3. Ben-Gurion University, Beersheba, Israel

    Meirav Zehavi

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  1. Sushmita Gupta
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  2. Sanjukta Roy
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  3. Saket Saurabh
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  4. Meirav Zehavi
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Correspondence to Sanjukta Roy.

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The preliminary version of this paper has appeared in MFCS 2017.

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Gupta, S., Roy, S., Saurabh, S. et al. Parameterized Algorithms and Kernels for Rainbow Matching. Algorithmica 81, 1684–1698 (2019). https://doi.org/10.1007/s00453-018-0497-3

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