Sensitivity Lower Bounds via Locally Testable Codes

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 $\Omega(n)$, strengthening the previous $\Omega(n^\delta)$ 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 $\Omega(k)$ 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 $\Omega(1/\varepsilon)$ 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-$\mathsf{LOCAL}$.