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Kernelization of Arc Disjoint Cycle Packing in α-Bounded Digraphs

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Abstract

In the Arc Disjoint Cycle Packing problem, we are given a simple directed graph (digraph) G, a positive integer k, and the task is to decide whether there exist k arc disjoint cycles. The problem is known to be W[1]-hard on general digraphs parameterized by the standard parameter k. In this paper we show that the problem admits a polynomial kernel on α-bounded digraphs. That is, we give a polynomial-time algorithm, that given an instance (D,k) of Arc Disjoint Cycle Packing, outputs an equivalent instance \((D^{\prime },k^{\prime })\) of Arc Disjoint Cycle Packing, such that \(k^{\prime }\leq k\) and the size of \(D^{\prime }\) is upper-bounded by a polynomial function of k. For any integer α ≥ 1, the class of α-bounded digraphs, denoted by \({\mathcal D}_{\alpha }\), contains a digraph D such that the maximum size of an independent set in D is at most α. That is, in D, any set of α + 1 vertices has an arc with both end-points in the set. For α = 1, this corresponds to the well-studied class of tournaments. Our results generalize the recent result by Bessy et al. [MFCS, 2019] about Arc Disjoint Cycle Packing on tournaments.

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Acknowledgements

We are grateful to William Lochet for the invaluable suggestions and discussions.

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Correspondence to Abhishek Sahu.

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This project received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant no. 819416), and the Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18.

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Sahu, A., Saurabh, S. Kernelization of Arc Disjoint Cycle Packing in α-Bounded Digraphs. Theory Comput Syst 67, 221–233 (2023). https://doi.org/10.1007/s00224-022-10114-8

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