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Set Cover Problems with Small Neighborhood Covers

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Abstract

In this paper, we study a class of set cover problems that satisfy a special property which we call the small neighborhood cover property. This class encompasses several well-studied problems including vertex cover, interval cover, bag interval cover and tree cover. We design unified sequential, parallel and distributed algorithms that can handle any set cover problem falling under the above framework and yield constant factor approximations. The algorithms run in NC in the parallel setting and can be executed in polylogarithmic communication rounds in the distributed setting.

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References

  1. Agarwal, A., Chakaravarthy, V., Choudhury, A., Roy, S., Sabharwal, Y.: Distributed and parallel algorithms for set cover problems with small neighborhood covers. In: 33rd Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), pp. 249–261 (2013)

  2. Bar-Noy, A., Bar-Yehuda, R., Freund, A., Naor, J., Schieber, B.: A unified approach to approximating resource allocation and scheduling. J. ACM 48(5), 1069–1090 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berman, P., DasGupta, B.: Improvements in throughout maximization for real-time scheduling. In: Proceedings of the 32nd ACM Symposium on Theory of Computing (STOC), pp. 680–687 (2000)

  4. Bertossi, A., Moretti, S.: Parallel algorithms on circular-arc graphs. Inf. Process. Lett. 33(6), 275–281 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chakaravarthy, V., Kumar, A., Roy, S., Sabharwal, Y.: Resource allocation for covering time varying demands. In: Proceedings of the 19th European Symposium on Algorithms (ESA), pp. 543–554 (2011)

  6. Chakrabarty, D., Grant, E., Könemann, J.: On column-restricted and priority covering integer programs. In: 14th International Conference on Integer Programming and Combinatorial Optimization (IPCO), pp. 365–368 (2010)

  7. Dinur, I., Guruswami, V., Khot, S., Regev, O.: A new multilayered PCP and the hardness of hypergraph vertex cover. SIAM J. Comput. 34(5), 1129–1146 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gandhi, R., Khuller, S., Srinivasan, A.: Approximation algorithms for partial covering problems. J. Algorithms 53(1), 55–84 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Grandoni, F., Könemann, J., Panconesi, A.: Distributed weighted vertex cover via maximal matchings. ACM Transactions on Algorithms 5(1), 839–848 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Khuller, S., Vishkin, U., Young, N.E.: A primal-dual parallel approximation technique applied to weighted set and vertex covers. J. Algorithms 17(2), 280–289 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  12. Koufogiannakis, C., Young, N.: Distributed algorithms for covering, packing and maximum weighted matching. Distrib. Comput. 24(1), 45–63 (2011)

    Article  MATH  Google Scholar 

  13. Kuhn, F., Moscibroda, T., Wattenhofer, R.: The price of being near-sighted. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 980–989 (2006)

  14. Linial, N., Saks, M.: Low diameter graph decompositions. Combinatorica 13(4), 441–454 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Luby, M.: A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15(4), 1036–1053 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  16. Panconesi, A., Sozio, M.: Fast primal-dual distributed algorithms for scheduling and matching problems. Distrib. Comput. 22(4), 269–283 (2010)

    Article  MATH  Google Scholar 

  17. Rajagopalan, S., Vazirani, V.: Primal-dual rnc approximation algorithms for set cover and covering integer programs. SIAM J. Comput. 28(2), 525–540 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  18. Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proceedings of the 29th ACM Symposium on Theory of Computing (STOC), pp. 475–484 (1997)

  19. Williamson, D., Shmoys, D.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  20. Ye, Y., Borodin, A.: Elimination graphs. ACM Trans. Algorithms 8(2), 14 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Brown University, Providence, RI, USA

    Archita Agarwal

  2. IBM Research, New Delhi, India

    Venkatesan T. Chakaravarthy, Anamitra R. Choudhury & Yogish Sabharwal

  3. Amazon, Bangalore, India

    Sambudha Roy

Authors
  1. Archita Agarwal
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  2. Venkatesan T. Chakaravarthy
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  3. Anamitra R. Choudhury
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  4. Sambudha Roy
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  5. Yogish Sabharwal
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Corresponding author

Correspondence to Venkatesan T. Chakaravarthy.

Additional information

A preliminary version of the paper appeared in the proceedings of FSTTCS 2013 [1]. Research conducted while Archita Agarwal and Sambudha Roy were at IBM Research, India.

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Agarwal, A., Chakaravarthy, V.T., Choudhury, A.R. et al. Set Cover Problems with Small Neighborhood Covers. Theory Comput Syst 62, 1763–1797 (2018). https://doi.org/10.1007/s00224-017-9842-1

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  • DOI: https://doi.org/10.1007/s00224-017-9842-1

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