Parameterized Inapproximability Hypothesis under ETH

Venkatesan Guruswami; Bingkai Lin; Xuandi Ren; Yican Sun; Kewen Wu

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arXiv · 2311.16587

Parameterized Inapproximability Hypothesis under ETH

Venkatesan Guruswami·Bingkai Lin·Xuandi Ren·Yican Sun·Kewen Wu

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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 $\varepsilon$ fraction of constraints for some absolute constant $\varepsilon > 0$. 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.

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BibTeX RIS

Venkatesan Guruswami, Bingkai Lin, Xuandi Ren, Yican Sun, Kewen Wu. 2023-11-28. Parameterized Inapproximability Hypothesis under ETH. https://arxiv.org/abs/2311.16587

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