Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sensitivity Lower Bounds for Approximaiton Algorithms

Noah Fleming, Yuichi Yoshida

TL;DR

This work establishes 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.

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.

Sensitivity Lower Bounds for Approximaiton Algorithms

TL;DR

This work establishes 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.

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.

Paper Structure

This paper contains 48 sections, 34 theorems, 76 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Sensitivity Lower Bounds
  4. Locality Lower Bounds for the Non-Signaling Model
  5. Proof Overview
  6. Degree Reduction.
  7. Expanderization.
  8. Gap amplification.
  9. Alphabet reduction.
  10. Discussions
  11. Parallel Repetition
  12. Stronger Sensitivity Lower Bounds.
  13. Serial Repetition.
  14. Hardness for Polynomial-time Tractable Problems.
  15. Average Sensitivity.
  16. ...and 33 more sections

Key Result

Theorem 1.1

There are universal constants $\varepsilon, \delta > 0$ such that any (inefficient) algorithm for the maximum clique problem that outputs an $n^{-\varepsilon}$-approximate clique with probability $1-O(1/n)$ has sensitivity $\Omega(n^\delta)$.

Figures (3)

  • Figure 1: DegreeReduction on a vertex $v$ with $d=4$. The intra-cloud edges represent an expander with $d_0=2$.
  • Figure 2: A graph $G$ before and after Expanderization. The edges of the expander are marked in red.
  • Figure 3: Conceptual illustration of the "Swiss cheese" $S \subseteq \{0,1\}^\ell$. Vertices of $\{0,1\}^\ell$ which are codewords are indicated by red circles, and the Hamming balls of radius $\ell/4$ around each one of them has been removed from $\{0,1\}^\ell$ to form $S$. The random threshold $\tau$ chosen in the recovery procedure is indicated by the dashed circle.

Theorems & Definitions (67)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: see, e.g., hoory2006expander
  • Lemma 2.2: see, e.g., hoory2006expander
  • Lemma 2.3
  • proof
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 57 more