Parameterized Inapproximability Hypothesis under ETH
Abstract
The Parameterized Inapproximability Hypothesis (PIH) asserts that no fixed parameter tractable (FPT) algorithm can distinguish a satisfiable CSP instance, parameterized by the number of variables, from one where every assignment fails to satisfy an fraction of constraints for some absolute constant . PIH plays the role of the PCP theorem in parameterized complexity. However, PIH has only been established under Gap-ETH, a very strong assumption with an inherent gap. In this work, we prove PIH under the Exponential Time Hypothesis (ETH). This is the first proof of PIH from a gap-free assumption. Our proof is self-contained and elementary. We identify an ETH-hard CSP whose variables take vector values, and constraints are either linear or of a special parallel structure. Both kinds of constraints can be checked with constant soundness via a "parallel PCP of proximity" based on the Walsh-Hadamard code.
Cite
@article{arxiv.2311.16587,
title = {Parameterized Inapproximability Hypothesis under ETH},
author = {Venkatesan Guruswami and Bingkai Lin and Xuandi Ren and Yican Sun and Kewen Wu},
journal= {arXiv preprint arXiv:2311.16587},
year = {2023}
}