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Scheme-Theoretic Approach to Computational Complexity. IV. A New Perspective on Hardness of Approximation

Ali Çivril

TL;DR

This paper advances a scheme-theoretic, unconditional framework for hardness of approximation, enabling direct, absolute super-polynomial time lower bounds without reliance on reductions or external hypotheses. It formalizes notions such as sub-problems, homogeneous and prime instances, and complexity measures, culminating in the Fundamental Lemma that ties instance structure to time complexity. The main results show exponential-time hardness for approximating MAX-$3$-SAT within a $7/8+\epsilon$ gap and MAX-$3$-LIN-2 within a $1/2+\epsilon$ gap, thereby supporting Gap-ETH and providing near-tight hardness in regimes where $\epsilon(n)$ can be very small. The approach relies on constructing prime, homogeneous simple sub-problems via combinatorial block techniques, offering a direct path from problem specification to hardness conclusions with potential broad impact on understanding approximability.

Abstract

We provide a new approach for establishing hardness of approximation results, based on the theory recently introduced by the author. It allows one to directly show that approximating a problem beyond a certain threshold requires super-polynomial time. To exhibit the framework, we revisit two famous problems in this paper. The particular results we prove are: MAX-3-SAT$(1,\frac{7}{8}+ε)$ requires exponential time for any constant $ε$ satisfying $\frac{1}{8} \geq ε> 0$. In particular, the gap exponential time hypothesis (Gap-ETH) holds. MAX-3-LIN-2$(1-ε, \frac{1}{2}+ε)$ requires exponential time for any constant $ε$ satisfying $\frac{1}{4} \geq ε> 0$.

Scheme-Theoretic Approach to Computational Complexity. IV. A New Perspective on Hardness of Approximation

TL;DR

This paper advances a scheme-theoretic, unconditional framework for hardness of approximation, enabling direct, absolute super-polynomial time lower bounds without reliance on reductions or external hypotheses. It formalizes notions such as sub-problems, homogeneous and prime instances, and complexity measures, culminating in the Fundamental Lemma that ties instance structure to time complexity. The main results show exponential-time hardness for approximating MAX--SAT within a gap and MAX--LIN-2 within a gap, thereby supporting Gap-ETH and providing near-tight hardness in regimes where can be very small. The approach relies on constructing prime, homogeneous simple sub-problems via combinatorial block techniques, offering a direct path from problem specification to hardness conclusions with potential broad impact on understanding approximability.

Abstract

We provide a new approach for establishing hardness of approximation results, based on the theory recently introduced by the author. It allows one to directly show that approximating a problem beyond a certain threshold requires super-polynomial time. To exhibit the framework, we revisit two famous problems in this paper. The particular results we prove are: MAX-3-SAT requires exponential time for any constant satisfying . In particular, the gap exponential time hypothesis (Gap-ETH) holds. MAX-3-LIN-2 requires exponential time for any constant satisfying .
Paper Structure (3 sections, 6 theorems, 2 equations, 6 tables)

This paper contains 3 sections, 6 theorems, 2 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of Theorem \ref{['3-sat']} and Theorem \ref{['3-lin-2']}

Key Result

Theorem 1

There exist infinitely many $n \in \mathbb{Z}^+$ such that for any constant $\delta > 0$, the problem $\mathsf{MAX\textrm{-}3\textrm{-}SAT}(1,\frac{7}{8}+\epsilon)$ cannot be solved deterministically in time less than $2^{(1-\delta)3 \epsilon n}$, where $n$ is the number of variables in the $\mathsf

Theorems & Definitions (9)

  • Theorem 1
  • Corollary 2
  • proof
  • Theorem 3
  • Lemma 4: Fundamental Lemma
  • Theorem 5
  • proof
  • Theorem 6
  • proof