The Parameterized Inapproximability Hypothesis (PIH), which is an analog of the PCP theorem in parameterized complexity, asserts that, there is a constant $\varepsilon> 0$ such that for any computable function $f:\mathbb{N}\to\mathbb{N}$, no $f(k)\cdot n^{O(1)}$-time algorithm can, on input a $k$-variable CSP instance with domain size $n$, find an assignment satisfying $1-\varepsilon$ fraction of the constraints. A recent work by Guruswami, Lin, Ren, Sun, and Wu (STOC'24) established PIH under the Exponential Time Hypothesis (ETH). In this work, we improve the quantitative aspects of PIH and prove (under ETH) that approximating sparse parameterized CSPs within a constant factor requires $n^{k^{1-o(1)}}$ time. This immediately implies that, assuming ETH, finding a $(k/2)$-clique in an $n$-vertex graph with a $k$-clique requires $n^{k^{1-o(1)}}$ time. We also prove almost optimal time lower bounds for approximating $k$-ExactCover and Max $k$-Coverage. Our proof follows the blueprint of the previous work to identify a "vector-structured" ETH-hard CSP whose satisfiability can be checked via an appropriate form of "parallel" PCP. Using further ideas in the reduction, we guarantee additional structures for constraints in the CSP. We then leverage this to design a parallel PCP of almost linear size based on Reed-Muller codes and derandomized low degree testing.

翻译：参数化不可近似性假说（PIH）是参数化复杂度中PCP定理的类比，其断言存在常数$\varepsilon>0$，使得对任意可计算函数$f:\mathbb{N}\to\mathbb{N}$，无$f(k)\cdot n^{O(1)}$时间算法能在输入域大小为$n$的$k$变量CSP实例时，找到满足$1-\varepsilon$比例约束的赋值。Guruswami、Lin、Ren、Sun和Wu（STOC'24）近期工作已在指数时间假说（ETH）下确立了PIH。本文改进了PIH的量化性质，并证明（在ETH下）在常数因子内逼近稀疏参数化CSP需要$n^{k^{1-o(1)}}$时间。这直接意味着，假设ETH成立，在包含$k$-团$n$个顶点的图中寻找$(k/2)$-团需要$n^{k^{1-o(1)}}$时间。我们还证明了逼近$k$-ExactCover和Max $k$-Coverage的几乎最优时间下界。证明沿袭先前工作框架，识别出一种“向量结构”的ETH困难CSP，其可满足性可通过适当形式的“并行”PCP检验。利用归约中的进一步思想，我们保证了CSP约束的附加结构，进而基于Reed-Muller码与去随机化低次测试，设计出几乎线性规模的并行PCP。