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On Equivalence of Parameterized Inapproximability of k-Median, k-Max-Coverage, and 2-CSP

Authors Karthik C. S. , Euiwoong Lee , Pasin Manurangsi

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LIPIcs.IPEC.2024.6.pdf
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ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → W hierarchy
Keywords
  • Parameterized complexity
  • Hardness of Approximation
  • Parameterized Inapproximability Hypothesis
  • max coverage
  • k-median

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Abstract

Parameterized Inapproximability Hypothesis (PIH) is a central question in the field of parameterized complexity. PIH asserts that given as input a 2-CSP on k variables and alphabet size n, it is 𝖶[1]-hard parameterized by k to distinguish if the input is perfectly satisfiable or if every assignment to the input violates 1% of the constraints. 
An important implication of PIH is that it yields the tight parameterized inapproximability of the k-maxcoverage problem. In the k-maxcoverage problem, we are given as input a set system, a threshold τ > 0, and a parameter k and the goal is to determine if there exist k sets in the input whose union is at least τ fraction of the entire universe. PIH is known to imply that it is 𝖶[1]-hard parameterized by k to distinguish if there are k input sets whose union is at least τ fraction of the universe or if the union of every k input sets is not much larger than τ⋅ (1-1/e) fraction of the universe. 
In this work we present a gap preserving FPT reduction (in the reverse direction) from the k-maxcoverage problem to the aforementioned 2-CSP problem, thus showing that the assertion that approximating the k-maxcoverage problem to some constant factor is 𝖶[1]-hard implies PIH. In addition, we present a gap preserving FPT reduction from the k-median problem (in general metrics) to the k-maxcoverage problem, further highlighting the power of gap preserving FPT reductions over classical gap preserving polynomial time reductions.

Cite As Get BibTex

Karthik C. S., Euiwoong Lee, and Pasin Manurangsi. On Equivalence of Parameterized Inapproximability of k-Median, k-Max-Coverage, and 2-CSP. In 19th International Symposium on Parameterized and Exact Computation (IPEC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 321, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.IPEC.2024.6

Author Details

Karthik C. S.
  • Rutgers University, Piscataway, NJ, USA
Euiwoong Lee
  • University of Michigan, Ann Arbor, MI, USA
Pasin Manurangsi
  • Google Research, Bangkok, Thailand

Funding

  • Karthik C. S.: This work was supported by the National Science Foundation under Grant CCF-2313372 and by the Simons Foundation, Grant Number 825876, Awardee Thu D. Nguyen.
  • Lee, Euiwoong: Supported in part by NSF grant CCF-2236669 and Google.
  • Manurangsi, Pasin: Part of this work was done while the author was visiting Rutgers University.

Acknowledgements

We are grateful to the Dagstuhl Seminar 23291 for a special collaboration opportunity. We thank Vincent Cohen-Addad, Venkatesan Guruswami, Jason Li, Bingkai Lin, and Xuandi Ren for discussions.

References

  1. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, 1998. URL: https://doi.org/10.1145/278298.278306.
  2. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of NP. J. ACM, 45(1):70-122, 1998. URL: https://doi.org/10.1145/273865.273901.
  3. Huck Bennett, Mahdi Cheraghchi, Venkatesan Guruswami, and João Ribeiro. Parameterized inapproximability of the minimum distance problem over all fields and the shortest vector problem in all lp norms. In Barna Saha and Rocco A. Servedio, editors, Proceedings of the 55th Annual ACM Symposium on Theory of Computing, STOC 2023, Orlando, FL, USA, June 20-23, 2023, pages 553-566. ACM, 2023. URL: https://doi.org/10.1145/3564246.3585214.
  4. Arnab Bhattacharyya, Édouard Bonnet, László Egri, Suprovat Ghoshal, Karthik C. S., Bingkai Lin, Pasin Manurangsi, and Dániel Marx. Parameterized intractability of even set and shortest vector problem. J. ACM, 68(3):16:1-16:40, 2021. URL: https://doi.org/10.1145/3444942.
  5. Arnab Bhattacharyya, Suprovat Ghoshal, Karthik C. S., and Pasin Manurangsi. Parameterized intractability of even set and shortest vector problem from gap-eth. In 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, July 9-13, 2018, Prague, Czech Republic, pages 17:1-17:15, 2018. URL: https://doi.org/10.4230/LIPIcs.ICALP.2018.17.
  6. Boris Bukh, Karthik C. S., and Bhargav Narayanan. Applications of random algebraic constructions to hardness of approximation. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 237-244. IEEE, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00032.
  7. Gruia Calinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing, 40(6):1740-1766, 2011. URL: https://doi.org/10.1137/080733991.
  8. Parinya Chalermsook, Marek Cygan, Guy Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, and Luca Trevisan. From gap-exponential time hypothesis to fixed parameter tractable inapproximability: Clique, dominating set, and more. SIAM J. Comput., 49(4):772-810, 2020. URL: https://doi.org/10.1137/18M1166869.
  9. Yijia Chen, Yi Feng, Bundit Laekhanukit, and Yanlin Liu. Simple combinatorial construction of the k^o(1)-lower bound for approximating the parameterized k-clique. CoRR, abs/2304.07516, 2023. URL: https://doi.org/10.48550/arXiv.2304.07516.
  10. Yijia Chen and Bingkai Lin. The constant inapproximability of the parameterized dominating set problem. SIAM J. Comput., 48(2):513-533, 2019. URL: https://doi.org/10.1137/17M1127211.
  11. Rajesh Chitnis, Andreas Emil Feldmann, and Pasin Manurangsi. Parameterized approximation algorithms for bidirected steiner network problems. ACM Transactions on Algorithms (TALG), 17(2):1-68, 2021. URL: https://doi.org/10.1145/3447584.
  12. Rajesh Chitnis, Andreas Emil Feldmann, and Ondrej Suchý. A tight lower bound for planar steiner orientation. Algorithmica, 81(8):3200-3216, 2019. URL: https://doi.org/10.1007/s00453-019-00580-x.
  13. Vincent Cohen-Addad, Anupam Gupta, Amit Kumar, Euiwoong Lee, and Jason Li. Tight fpt approximations for k-median and k-means. arXiv preprint, 2019. URL: https://arxiv.org/abs/1904.12334.
  14. Vincent Cohen-Addad, Anupam Gupta, Amit Kumar, Euiwoong Lee, and Jason Li. Tight FPT approximations for k-median and k-means. In Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, and Stefano Leonardi, editors, 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, volume 132 of LIPIcs, pages 42:1-42:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.42.
  15. Vincent Cohen-Addad Viallat, Fabrizio Grandoni, Euiwoong Lee, and Chris Schwiegelshohn. Breaching the 2 lmp approximation barrier for facility location with applications to k-median. In Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 940-986. SIAM, 2023. Google Scholar
  16. Pierluigi Crescenzi, Riccardo Silvestri, and Luca Trevisan. On weighted vs unweighted versions of combinatorial optimization problems. Inf. Comput., 167(1):10-26, 2001. URL: https://doi.org/10.1006/inco.2000.3011.
  17. Irit Dinur. The PCP theorem by gap amplification. J. ACM, 54(3):12, 2007. URL: https://doi.org/10.1145/1236457.1236459.
  18. Irit Dinur. Mildly exponential reduction from gap 3sat to polynomial-gap label-cover. Electronic Colloquium on Computational Complexity (ECCC), 23:128, 2016. URL: http://eccc.hpi-web.de/report/2016/128, URL: https://arxiv.org/abs/TR16-128.
  19. Pavel Dvořák, Andreas Emil Feldmann, Ashutosh Rai, and Paweł Rzażewski. Parameterized inapproximability of independent set in h-free graphs. Algorithmica, pages 1-27, 2022. Google Scholar
  20. Uriel Feige. A threshold of ln n for approximating set cover. Journal of the ACM (JACM), 45(4):634-652, 1998. URL: https://doi.org/10.1145/285055.285059.
  21. Uriel Feige, Shafi Goldwasser, László Lovász, Shmuel Safra, and Mario Szegedy. Interactive proofs and the hardness of approximating cliques. J. ACM, 43(2):268-292, 1996. URL: https://doi.org/10.1145/226643.226652.
  22. Andreas Emil Feldmann, Karthik C. S., Euiwoong Lee, and Pasin Manurangsi. A survey on approximation in parameterized complexity: Hardness and algorithms. Algorithms, 13(6):146, 2020. URL: https://doi.org/10.3390/a13060146.
  23. Sudipto Guha and Samir Khuller. Greedy strikes back: Improved facility location algorithms. Journal of algorithms, 31(1):228-248, 1999. URL: https://doi.org/10.1006/JAGM.1998.0993.
  24. Venkatesan Guruswami, Bingkai Lin, Xuandi Ren, Yican Sun, and Kewen Wu. Almost optimal time lower bound for approximating parameterized clique, csp, and more, under ETH. CoRR, abs/2404.08870, 2024. URL: https://doi.org/10.48550/arXiv.2404.08870.
  25. Venkatesan Guruswami, Bingkai Lin, Xuandi Ren, Yican Sun, and Kewen Wu. Parameterized inapproximability hypothesis under ETH. In STOC, 2024. Google Scholar
  26. Dorit S Hochba. Approximation algorithms for np-hard problems. ACM Sigact News, 28(2):40-52, 1997. URL: https://doi.org/10.1145/261342.571216.
  27. Russell Impagliazzo and Ramamohan Paturi. On the complexity of k-SAT. J. Comput. Syst. Sci., 62(2):367-375, 2001. URL: https://doi.org/10.1006/jcss.2000.1727.
  28. Russell Impagliazzo, Ramamohan Paturi, and Francis Zane. Which problems have strongly exponential complexity? J. Comput. Syst. Sci., 63(4):512-530, 2001. URL: https://doi.org/10.1006/jcss.2001.1774.
  29. Tapas Kanungo, David M Mount, Nathan S Netanyahu, Christine D Piatko, Ruth Silverman, and Angela Y Wu. A local search approximation algorithm for k-means clustering. Computational Geometry, 28(2-3):89-112, 2004. URL: https://doi.org/10.1016/J.COMGEO.2004.03.003.
  30. Karthik C. S. and Subhash Khot. Almost polynomial factor inapproximability for parameterized k-clique. In Shachar Lovett, editor, 37th Computational Complexity Conference, CCC 2022, July 20-23, 2022, Philadelphia, PA, USA, volume 234 of LIPIcs, pages 6:1-6:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.CCC.2022.6.
  31. Karthik C. S., Bundit Laekhanukit, and Pasin Manurangsi. On the parameterized complexity of approximating dominating set. J. ACM, 66(5):33:1-33:38, 2019. URL: https://doi.org/10.1145/3325116.
  32. Karthik C. S. and Pasin Manurangsi. On closest pair in euclidean metric: Monochromatic is as hard as bichromatic. Combinatorica, 40(4):539-573, 2020. URL: https://doi.org/10.1007/S00493-019-4113-1.
  33. Karthik C. S. and Inbal Livni Navon. On hardness of approximation of parameterized set cover and label cover: Threshold graphs from error correcting codes. In Hung Viet Le and Valerie King, editors, 4th Symposium on Simplicity in Algorithms, SOSA 2021, Virtual Conference, January 11-12, 2021, pages 210-223. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976496.24.
  34. Subhash Khot. On the power of unique 2-prover 1-round games. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 767-775. ACM, 2002. URL: https://doi.org/10.1145/509907.510017.
  35. Subhash Khot, Dor Minzer, and Muli Safra. On independent sets, 2-to-2 games, and grassmann graphs. In Hamed Hatami, Pierre McKenzie, and Valerie King, editors, Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2017, Montreal, QC, Canada, June 19-23, 2017, pages 576-589. ACM, 2017. URL: https://doi.org/10.1145/3055399.3055432.
  36. Subhash Khot, Dor Minzer, and Muli Safra. Pseudorandom sets in grassmann graph have near-perfect expansion. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 592-601. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00062.
  37. Vladimir Kolmogorov, Andrei A. Krokhin, and Michal Rolínek. The complexity of general-valued csps. In Venkatesan Guruswami, editor, IEEE 56th Annual Symposium on Foundations of Computer Science, FOCS 2015, Berkeley, CA, USA, 17-20 October, 2015, pages 1246-1258. IEEE Computer Society, 2015. URL: https://doi.org/10.1109/FOCS.2015.80.
  38. Bingkai Lin. The parameterized complexity of the k-biclique problem. J. ACM, 65(5):34:1-34:23, 2018. URL: https://doi.org/10.1145/3212622.
  39. Bingkai Lin. A simple gap-producing reduction for the parameterized set cover problem. In 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019, July 9-12, 2019, Patras, Greece, pages 81:1-81:15, 2019. URL: https://doi.org/10.4230/LIPIcs.ICALP.2019.81.
  40. Bingkai Lin. Constant approximating k-clique is w [1]-hard. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1749-1756, 2021. URL: https://doi.org/10.1145/3406325.3451016.
  41. Bingkai Lin, Xuandi Ren, Yican Sun, and Xiuhan Wang. Improved hardness of approximating k-clique under ETH. In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 285-306. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00025.
  42. Bingkai Lin, Xuandi Ren, Yican Sun, and Xiuhan Wang. Constant approximating parameterized k-setcover is W[2]-hard. In Nikhil Bansal and Viswanath Nagarajan, editors, Proceedings of the 2023 ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, Florence, Italy, January 22-25, 2023, pages 3305-3316. SIAM, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH126.
  43. Daniel Lokshtanov, M. S. Ramanujan, Saket Saurabh, and Meirav Zehavi. Parameterized complexity and approximability of directed odd cycle transversal. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 2181-2200, 2020. URL: https://doi.org/10.1137/1.9781611975994.134.
  44. Pasin Manurangsi. Tight running time lower bounds for strong inapproximability of maximum k-coverage, unique set cover and related problems (via t-wise agreement testing theorem). In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 62-81. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.5.
  45. Pasin Manurangsi and Prasad Raghavendra. A birthday repetition theorem and complexity of approximating dense csps. CoRR, abs/1607.02986, 2016. URL: https://arxiv.org/abs/1607.02986.
  46. Naoto Ohsaka. On the parameterized intractability of determinant maximization. arXiv preprint, 2022. URL: https://doi.org/10.48550/arXiv.2209.12519.
  47. Michal Wlodarczyk. Inapproximability within W[1]: the case of steiner orientation. CoRR, abs/1907.06529, 2019. URL: https://arxiv.org/abs/1907.06529.
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