research-article

Iterative Partial Rounding for Vertex Cover with Hard Capacities

Author:

Mong-Jen KaoAuthors Info & Claims

Algorithmica, Volume 83, Issue 1

Pages 45 - 71

https://doi.org/10.1007/s00453-020-00749-9

Published: 01 January 2021 Publication History

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.

References

[1]

An H-C, Bhaskara A, Chekuri C, Gupta S, Madan V, and Svensson O Centrality of trees for capacitated

k

-center Math. Program. 2015 154 1–2 29-53

[2]

An H-C, Singh M, and Svensson O LP-based algorithms for capacitated facility location SIAM J. Comput. 2017 46 1 272-306

[3]

Bansal, M., Garg, N., Gupta, N.: A 5-approximation for capacitated facility location. In: ESA’12, pp. 133–144 (2012)

[4]

Bar-Yehuda R and Even S A linear-time approximation algorithm for the weighted vertex cover problem J. Algorithms 1981 2 2 198-203

[5]

Byrka J, Pensyl T, Rybicki B, Srinivasan A, and Trinh K An improved approximation for

k

-median, and positive correlation in budgeted optimization ACM Trans. Algorithms 2017 13 2 23:1-23:31

[6]

Cheung, W.-C., Goemans, M., Wong, S.: Improved algorithms for vertex cover with hard capacities on multigraphs and hypergraphs. In: SODA’14 (2014)

[7]

Chuzhoy, J., Naor, J.: Covering problems with hard capacities. In: Proceedings of the 43rd Symposium on Foundations of Computer Science (FOCS) 2002, 16–19 November 2002, Vancouver, BC, Canada, pp. 481–489. IEEE Computer Society (2002).

[8]

Chuzhoy J and Naor J Covering problems with hard capacities SIAM J. Comput. 2006 36 2 498-515

[9]

Cygan M, Pilipczuk M, and Wojtaszczyk JO Capacitated domination faster than O(

2^{n}

) Inf. Process. Lett 2011 111 (23–24) 1099-1103

[10]

Dom, M., Lokshtanov, D., Saurabh, S., Villanger, Y.: Capacitated domination and covering: a parameterized perspective. In: IWPEC’08, pp. 78–90 (2008)

[11]

Gandhi R, Halperin E, Khuller S, Kortsarz G, and Srinivasan A An improved approximation algorithm for vertex cover with hard capacities J. Comput. Syst. Sci. 2006 72 16-33

[12]

Gandhi R, Khuller S, Parthasarathy S, and Srinivasan A Dependent rounding and its applications to approximation algorithms J. ACM 2006 53 3 324-360

[13]

Gandhi R, Khuller S, and Srinivasan A Approximation algorithms for partial covering problems J. Algorithms 2004 53 1 55-84

[14]

Grandoni F, Könemann J, Panconesi A, and Sozio M A primal-dual bicriteria distributed algorithm for capacitated vertex cover SIAM J. Comput. 2008 38 3 825-840

[15]

Guha S, Hassin R, Khuller S, and Or E Capacitated vertex covering J. Algorithms 2003 48 1 257-270

[16]

Guha S and Khuller S Greedy strikes back: Improved facility location algorithms J. Algorithms 1999 31 1 228-248

[17]

Hochbaum DS Approximation algorithms for the set covering and vertex cover problems SIAM J. Comput. 1982 11 3 555-556

[18]

Hochbaum DS and Shmoys DB A best possible heuristic for the

k

-center problem Math. Oper. Res. 1985 10 2 180-184

[19]

Kao M-J, Chen H-L, and Lee DT Capacitated domination: problem complexity and approximation algorithms Algorithmica 2015 72 1-43

[20]

Kao M-J, Liao C-S, and Lee DT Algorithmica Capacitated domination problem 2011 60 2 274-300

[21]

Kao M-J, Tu H-L, and Lee DT An O(

f

) bi-approximation for weighted capacitated covering with hard capacities Algorithmica 2019 81 5 1800-1817

[22]

Khot S and Regev O Vertex cover might be hard to approximate to within

2 - ϵ

J. Comput. Syst. Sci. 2008 74 3 335-349

[23]

Li S A 1.488 approximation algorithm for the uncapacitated facility location problem Inf. Comput. 2013 222 45-58

[24]

Li S On uniform capacitated

k

-median beyond the natural lp relaxation ACM Trans. Algorithms 2017 13 2 22:1-22:18

[25]

Li S and Svensson O Approximating

k

-median via pseudo-approximation SIAM J. Comput. 2016 45 2 530-547

[26]

Liedloff M, Todinca I, and Villanger Y Solving capacitated dominating set by using covering by subsets and maximum matching Discret. Appl. Math. 2014 168 60-68

[27]

Saha, B., Khuller, S.: Set cover revisited: hypergraph cover with hard capacities. In: ICALP’12, pp. 762–773 (2012)

[28]

Schrijver A Theory of Linear and Integer Programming 1986 New York Wiley

[29]

Wolsey LA An analysis of the greedy algorithm for the submodular set covering problem Combinatorica 1982 2 4 385-393

Cited By

Zhang QLi WSu QZhang X(2022)A Local-Ratio-Based Power Control Approach for Capacitated Access Points in Mobile Edge ComputingProceedings of the 6th International Conference on High Performance Compilation, Computing and Communications10.1145/3546000.3546027(175-182)Online publication date: 23-Jun-2022
https://dl.acm.org/doi/10.1145/3546000.3546027

Index Terms

Iterative Partial Rounding for Vertex Cover with Hard Capacities
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
  2. Mathematical analysis
2. Theory of computation
  1. Design and analysis of algorithms
    1. Approximation algorithms analysis

Index terms have been assigned to the content through auto-classification.

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Published In

cover image Algorithmica

Algorithmica Volume 83, Issue 1

Jan 2021

412 pages

ISSN:0178-4617

Issue’s Table of Contents

© Springer Science+Business Media, LLC, part of Springer Nature 2020.

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 January 2021

Accepted: 15 July 2020

Received: 07 August 2017

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Ministry of Science and Technology, Taiwan

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Citations

Cited By

Zhang QLi WSu QZhang X(2022)A Local-Ratio-Based Power Control Approach for Capacitated Access Points in Mobile Edge ComputingProceedings of the 6th International Conference on High Performance Compilation, Computing and Communications10.1145/3546000.3546027(175-182)Online publication date: 23-Jun-2022
https://dl.acm.org/doi/10.1145/3546000.3546027

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