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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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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.

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