Metric Sublinear Algorithms via Linear Sampling

TL;DR

This paper introduces a linear sampling technique for metric graphs that enables sublinear time approximation algorithms for problems like densest subgraph and max cut, significantly improving efficiency.

Contribution

The authors develop a novel sampling method that sparsifies metric graphs using few edge queries, leading to faster approximation algorithms for key graph problems.

Findings

First sublinear time algorithm for densest subgraph in metric spaces

Improved running time for average distance estimation

Effective graph sparsification preserves approximation quality

Abstract

In this work we provide a new technique to design fast approximation algorithms for graph problems where the points of the graph lie in a metric space. Specifically, we present a sampling approach for such metric graphs that, using a sublinear number of edge weight queries, provides a {\em linear sampling}, where each edge is (roughly speaking) sampled proportionally to its weight. For several natural problems, such as densest subgraph and max cut among others, we show that by sparsifying the graph using this sampling process, we can run a suitable approximation algorithm on the sparsified graph and the result remains a good approximation for the original problem. Our results have several interesting implications, such as providing the first sublinear time approximation algorithm for densest subgraph in a metric space, and improving the running time of estimating the average distance.