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Iterative Partial Rounding for Vertex Cover with Hard Capacities

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Abstract

We provide a simple and novel algorithmic design technique, for which we call iterative partial rounding, that gives a tight rounding-based approximation for vertex cover with hard capacities (VC-HC). In particular, we obtain an f-approximation for VC-HC on hypergraphs, improving over a previous results of Cheung et al. (In: SODA’14, 2014) to the tight extent. This also closes the gap of approximation since it was posted by Chuzhoy and Naor (Proceedings of the 43rd Symposium on Foundations of Computer Science (FOCS) 2002, pp. 481--489. IEEE Computer Society, 2002). Our main technical tool for establishing the approximation guarantee is a separation lemma that certifies the existence of a strong partition for solutions that are basic feasible in an extended version of the natural LP. We believe that our rounding technique is of independent interest when hard constraints are considered.

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Notes

  1. An augmented \((\alpha ,\beta )\)-covering is a solution that violates the multiplicity limit by a factor \(\alpha\) and has a cost factor \(\beta\) to the optimal LP solution for VC-HC.

  2. Recall that \(p \in {\mathbf {Q}}(\Psi )\) is an extreme point if it is not in the interior of any line segment contained in \({\mathbf {Q}}(\Psi )\), i.e., \(p = \lambda r + (1-\lambda ) s\) for some \(0< \lambda < 1\) implies that either \(r \notin {\mathbf {Q}}(\Psi )\) or \(s \notin {\mathbf {Q}}(\Psi )\).

  3. Note that, in Equation (6) we also drop out \(h_{e,v}\) for all \(e \in {\mathcal C}^{(E)}\) and \(v \in e {\setminus } e^{{\text {actv}}}_h\) since by definition they are zero at the considered extreme point p.

  4. If not, the column reduction (Gaussian elimination on the columns) would have led to a rank less than \(\left| X \cup H^*\right|\), a contradiction to the fact that \(\tilde{M}\) has a full column rank.

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Acknowledgements

The author would like to thank Kai-Min Chung, Herbert Yu, and the anonymous reviewers for their valuable comments on the presentation of this work.

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  1. Department of Computer Science and Information Engineering, National Chung-Cheng University, Chiayi, Taiwan

    Mong-Jen Kao

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An extended abstract of this work appeared in the ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), Jan 16–19, Barcelona, Spain. This work is supported in part by the Ministry of Science and Technology (MOST), Taiwan, Under Grants MOST107-2218-E-194-015-MY3 and MOST108-2221-E-194-026-MY3.

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Kao, MJ. Iterative Partial Rounding for Vertex Cover with Hard Capacities. Algorithmica 83, 45–71 (2021). https://doi.org/10.1007/s00453-020-00749-9

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