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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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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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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

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.

Subject Classification

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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