Almost Optimal Time Lower Bound for Approximating Parameterized Clique, CSP, and More, under ETH

TL;DR

The paper advances the fine-grained understanding of parameterized hardness by proving that, under ETH, approximating sparse parameterized CSPs within a constant factor requires time $f(k) n^{k^{1-o(1)}}$, implying near-optimal ETH-based lower bounds for problems such as $k$-Clique. It achieves this through a refined reduction chain that produces a special vector-valued CSP (SVecCSP) with structured parallel and linear constraints, enabling an almost-linear size probabilistic proof via a parallel Reed-Muller encoding and derandomized low-degree testing. A new PCP characterization for 3SAT emerges as a byproduct, linking parameterized complexity to classical PCP frameworks with a smooth trade-off between proof length and alphabet size. The work also yields near-optimal ETH-based hardness for $k$-ExactCover and Max $k$-Coverage, and outlines implications for other canonical parameterized problems, highlighting a path toward linear-size PCPs in the parameterized regime. Overall, the results substantially tighten the landscape of approximability lower bounds under ETH and introduce a powerful, scalable PCP toolkit for parameterized problems.

Abstract

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.

Figures (2)

• Figure 1: An example of $G'$ and $G"$ and the permutation to parallelize sub-constraints.
• Figure 2: An illustration of the reduction from $G$ to $\widehat{G}$, and from $\widehat{G}$ to $G'$.

Theorems & Definitions (84)

• Theorem 1.1
• Theorem 1.2: Informal version of \ref{['thm:main']}
• Corollary 1.3
• Theorem 1.4
• Proposition 1.5
• Theorem 1.6
• Proposition 1.7
• Theorem 1.8: Informal Version of \ref{['thm:new-pcp-restate']}
• Definition 3.1: 4-Regular 3-Coloring
• Theorem 3.2: ETH Lower Bound for 4-Regular 3-Coloring CFG+16