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.

翻译：参数化不可近似性假设（PIH）断言：不存在固定参数可解（FPT）算法能够区分一个可满足的CSP实例（以变量数量为参数化指标）与另一个在任意赋值下都无法满足绝对常数ε>0比例约束的实例。PIH在参数化复杂度中扮演着PCP定理的角色。然而，PIH此前仅能在Gap-ETH（一种具有固有间隙的极强假设）下得到证明。本文在指数时间假设（ETH）下证明了PIH。这是首次基于无间隙假设对PIH进行的证明。我们的证明是自洽且初等的。我们识别出一个ETH-hard的CSP问题，其变量取向量值，约束条件要么是线性约束，要么是特殊的并行结构约束。通过基于Walsh-Hadamard码的“并行邻近PCP”，这两类约束均能以常数的可靠性进行验证。