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The Parameterized Complexity of Cycle Packing: Indifference is Not an Issue

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Abstract

In the Cycle Packing problem, we are given an undirected graph G, a positive integer r, and the task is to check whether there exist r vertex-disjoint cycles. In this paper, we study Cycle Packing with respect to a structural parameter, namely, distance to proper interval graphs (indifference graphs). In particular, we show that Cycle Packing is fixed-parameter tractable (FPT) when parameterized by t, the size of a proper interval deletion set. For this purpose, we design an algorithm with \(\mathcal {O}(2^{\mathcal {O}(t \log t)} n^{\mathcal {O}(1)})\) running time. Bodlaender et al. (Theor Comput Sci 511:117–136, 2013) studied several structural parameterizations for Cycle Packing and our FPT algorithm fills a gap in their ecology of parameterizations. We combine color coding, greedy strategy and dynamic programming based on structural properties of proper interval graphs in a non-trivial fashion to obtain the FPT algorithm. Our belief is that this approach is quite general and can be useful in solving many other problems with the same parameterization.

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Notes

  1. The algorithm described in [26] does not seem to be correct as it does not produce the correct answer for a family of input instances.

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Authors and Affiliations

  1. Indian Institute of Technology, Palakkad, India

    R. Krithika

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

    Abhishek Sahu & Saket Saurabh

  3. University of Bergen, Bergen, Norway

    Saket Saurabh

  4. Ben-Gurion University, Beersheba, Israel

    Meirav Zehavi

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  1. R. Krithika
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  2. Abhishek Sahu
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  3. Saket Saurabh
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  4. Meirav Zehavi
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A preliminary version of this work appears in the proceedings of the 13th Latin American Theoretical INformatics Symposium (LATIN 2018).

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Krithika, R., Sahu, A., Saurabh, S. et al. The Parameterized Complexity of Cycle Packing: Indifference is Not an Issue. Algorithmica 81, 3803–3841 (2019). https://doi.org/10.1007/s00453-019-00599-0

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