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Scheme-Theoretic Approach to Computational Complexity. III. SETH

Ali Çivril

TL;DR

The paper proves a SETH-style deterministic lower bound for solving $k$-SAT with $k \ge 3$ by constructing a prime homogeneous simple sub-problem that encodes a large number of instances. It then mixes blocks to enforce primeness while preserving the Hilbert-polynomial structure, enabling a counting argument. A Hilbert-polynomial and Stirling-based analysis yields a lower bound of $\tau(k\text{-SAT}(n)) \ge (2^{k-\frac{3}{2}-\epsilon})^{\frac{n}{k+1}}$ for infinitely many $n$ and any $\epsilon>0$, with $n$ the number of variables. This strengthens unconditional hardness results for exact $k$-SAT under SETH and links algebraic considerations to deterministic time complexity.

Abstract

We show that there exist infinitely many $n \in \mathbb{Z}^+$ such that for any constant $ε> 0$, any deterministic algorithm to solve $k$-\textsf{SAT} for $k \geq 3$ must perform at least $(2^{k-\frac{3}{2}-ε})^{\frac{n}{k+1}}$ operations, where $n$ is the number of variables in the $k$\textsf{-SAT} instance.

Scheme-Theoretic Approach to Computational Complexity. III. SETH

TL;DR

The paper proves a SETH-style deterministic lower bound for solving -SAT with by constructing a prime homogeneous simple sub-problem that encodes a large number of instances. It then mixes blocks to enforce primeness while preserving the Hilbert-polynomial structure, enabling a counting argument. A Hilbert-polynomial and Stirling-based analysis yields a lower bound of for infinitely many and any , with the number of variables. This strengthens unconditional hardness results for exact -SAT under SETH and links algebraic considerations to deterministic time complexity.

Abstract

We show that there exist infinitely many such that for any constant , any deterministic algorithm to solve -\textsf{SAT} for must perform at least operations, where is the number of variables in the \textsf{-SAT} instance.
Paper Structure (2 sections, 3 theorems, 19 equations)

This paper contains 2 sections, 3 theorems, 19 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['main']}

Key Result

Theorem 1

For any integer $k \geq 3$, there exist infinitely many $n \in \mathbb{Z}^+$ such that for any constant $\epsilon > 0$, any deterministic algorithm must perform at least operations to solve the problem $k$-SAT, where $n$ is the number of variables in the $k$-SAT instance.

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 2: Fundamental Lemma
  • Theorem 3
  • proof