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Baby PIH: Parameterized Inapproximability of Min CSP

Authors Venkatesan Guruswami , Xuandi Ren , Sai Sandeep

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Author Details

Venkatesan Guruswami
  • Simons Institute for the Theory of Computing, Berkeley, CA, USA
  • Departments of EECS and Mathematics, University of California, Berkeley, CA, USA
Xuandi Ren
  • Department of EECS, University of California, Berkeley, CA, USA
Sai Sandeep
  • Department of EECS, University of California, Berkeley, CA, USA

Funding

  • Guruswami, Venkatesan: Research supported in part by NSF grants CCF-2228287 and CCF-2211972 and a Simons Investigator award.
  • Ren, Xuandi: Supported in part by NSF CCF-2228287.
  • Sandeep, Sai: Supported in part by NSF CCF-2228287.

Cite AsGet BibTex

Venkatesan Guruswami, Xuandi Ren, and Sai Sandeep. Baby PIH: Parameterized Inapproximability of Min CSP. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.CCC.2024.27

Abstract

The Parameterized Inapproximability Hypothesis (PIH) is the analog of the PCP theorem in the world of parameterized complexity. It asserts that no FPT algorithm can distinguish a satisfiable 2CSP instance from one which is only (1-ε)-satisfiable (where the parameter is the number of variables) for some constant 0 < ε < 1. We consider a minimization version of CSPs (Min-CSP), where one may assign r values to each variable, and the goal is to ensure that every constraint is satisfied by some choice among the r × r pairs of values assigned to its variables (call such a CSP instance r-list-satisfiable). We prove the following strong parameterized inapproximability for Min CSP: For every r ≥ 1, it is W[1]-hard to tell if a 2CSP instance is satisfiable or is not even r-list-satisfiable. We refer to this statement as "Baby PIH", following the recently proved Baby PCP Theorem (Barto and Kozik, 2021). Our proof adapts the combinatorial arguments underlying the Baby PCP theorem, overcoming some basic obstacles that arise in the parameterized setting. Furthermore, our reduction runs in time polynomially bounded in both the number of variables and the alphabet size, and thus implies the Baby PCP theorem as well.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Parameterized Inapproximability Hypothesis
  • Constraint Satisfaction Problems

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