Sensitivity Lower Bounds via Locally Testable Codes

2026-06-30

AI Summary

This study investigates the fundamental limits of output stability for approximation algorithms under infinitesimal input perturbations, formalized as sensitivity lower bounds. By transforming locally testable codes (LTCs) into constraint satisfaction problems (CSPs), the work introduces a general framework that, for the first time, leverages LTCs to establish linear sensitivity lower bounds for classic problems such as Max E3LIN2, and extends these results to settings like bipartite Max Cut. Combining complexity-theoretic reductions, analysis of Boolean function influences, and locality-preserving transformations from non-signaling models, the paper demonstrates that the sensitivity lower bounds for Max E3LIN2, Maximum Clique, and k-Cover tightly match their trivial upper bounds. This reveals an intrinsic barrier: even arbitrarily small input perturbations can induce substantial changes in near-optimal solutions.

Abstract

Sensitivity quantifies how far an algorithm's output can move in Hamming distance when a single input element is perturbed. We present a general scheme turning any locally testable code (LTC) into a sensitivity lower bound for the CSP encoded by the code. Instantiating it with the $c^3$-LTC yields a constant $\varepsilon > 0$ for which every $(1-\varepsilon)$-approximation algorithm for satisfiable Max E3LIN2 has sensitivity $Ω(n)$, strengthening the previous $Ω(n^δ)$ lower bound known only for general instances. Standard reductions give an $n^{1-O(\varepsilon)}$ lower bound for $n^{-\varepsilon}$-approximation algorithms for maximum clique, and an $Ω(k)$ bound for $(1-\varepsilon)$-approximation algorithms for maximum $k$-coverage, among others. These lower bounds match trivial upper bounds and imply that even slightly stable algorithms cannot achieve these approximations. A second instantiation, using a simple repetition code, shows that any $(1-\varepsilon)$-approximation algorithm for Max Cut on bipartite graphs has sensitivity $Ω(1/\varepsilon)$ as long as $\varepsilon = O(1/\sqrt{n})$, extending the prior result for exact algorithms (the regime $\varepsilon < 1/n$). Thus even on bipartite graphs, where a perfect cut always exists, near-optimal solutions cannot be maintained stably. Our sensitivity lower bounds have two applications. First, they yield averaged sensitivity lower bounds, implying that any nearly optimal randomized algorithm for Max E3SAT has linearly many output bits that, after fixing the random seed, are Boolean functions with large influence. Second, via the sensitivity-to-locality connection, they imply locality lower bounds in the non-signaling model, which generalizes $\mathsf{LOCAL}$ and quantum-$\mathsf{LOCAL}$.

Problem

• sensitivity
• approximation algorithms
• locally testable codes
• lower bounds
• stability

Innovation

• sensitivity lower bounds
• locally testable codes
• approximation algorithms
• non-signaling locality
• constraint satisfaction problems