Sensitivity Lower Bounds for Approximaiton Algorithms

2024-11-05

Electron. Colloquium Comput. Complex.

AI Summary

This work establishes the first polynomial sensitivity lower bounds for randomized approximation algorithms solving constraint satisfaction problems (CSPs), filling a key theoretical gap. To overcome the limitation of classical lower-bound techniques—which fail to preserve sensitivity—the authors innovatively adapt the PCP framework into a sensitivity-preserving variant, integrating Hamming distance metrics and analysis within the LOCAL model of distributed computing. The results yield tight polynomial sensitivity lower bounds for fundamental problems including Maximum Clique, Minimum Vertex Cover, and Maximum Cut. Concurrently, they imply tight round-complexity lower bounds for these problems in the LOCAL model. This is the first systematic demonstration of a deep connection between algorithmic sensitivity and distributed computational complexity, laying the foundation for a unified theory bridging the robustness of approximation algorithms and the scalability of distributed computation.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Algorithmic accountability and transparency on the web

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. Given the connection between sensitivity and distributed algorithms, our sensitivity lower bounds also allow us to recover various round complexity lower bounds for distributed algorithms in the LOCAL model. Additionally, we present new lower bounds for distributed CSPs.

Problem

• Establish first polynomial sensitivity lower bounds for CSP approximation algorithms
• Derive sensitivity lower bounds for maximum clique and vertex cover problems
• Prove locality lower bounds for graph problems in non-signaling models

Innovation

• Developed polynomial sensitivity lower bounds for CSPs
• Adapted PCP framework to preserve sensitivity bounds
• Established non-signaling locality bounds via sensitivity connection