Sensitivity Lower Bounds for Approximaiton Algorithms

TL;DR

This paper establishes the first polynomial lower bounds on the sensitivity of approximation algorithms for CSPs and related problems, using a novel adaptation of the PCP framework to sensitivity analysis.

Contribution

It introduces the first sensitivity lower bounds for approximation algorithms for CSPs by adapting the PCP framework, and connects sensitivity to locality in the non-signaling model.

Findings

Polynomial sensitivity lower bounds for CSP approximation algorithms.

Sensitivity bounds for maximum clique, vertex cover, and max cut.

Locality lower bounds in the non-signaling model for graph problems.

Abstract

Sensitivity measures how much the output of an algorithm changes, in terms of Hamming distance, when part of the input is modified. While approximation algorithms with low sensitivity have been developed for many problems, no sensitivity lower bounds were previously known for approximation algorithms. In this work, we establish the first polynomial lower bound on the sensitivity of (randomized) approximation algorithms for constraint satisfaction problems (CSPs) by adapting the probabilistically checkable proof (PCP) framework to preserve sensitivity lower bounds. From this, we derive polynomial sensitivity lower bounds for approximation algorithms for a variety of problems, including maximum clique, minimum vertex cover, and maximum cut. Leveraging the connection between sensitivity and locality in the non-signaling model, which subsumes the LOCAL, quantum-LOCAL, and bounded dependence models, we establish locality lower bounds for several graph problems in the non-signaling model.