2024/04/13 by Venkatesan Guruswami, Guruswami, Venkatesan, Bingkai Lin +7 · 1 voice · 1 citation
Computer Science · #Advanced Graph Theory Research #Algorithms and Data Compression #Complexity and Algorithms in Graphs #Computational Complexity (cs.CC) #FOS: Computer and information sciences
paper · pdf · doi:10.48550/arxiv.2404.08870
openalex publication_date 2024/04/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Parameterized Inapproximability Hypothesis (PIH), which is an analog of the PCP theorem in parameterized complexity, asserts that, there is a constant ε> 0 such that for any computable function f:ℕ→ℕ, no f(k)⋅ nO(1)-time algorithm can, on input a k-variable CSP instance with domain size n, find an assignment satisfying 1-ε 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^k1-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^k1-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.