The Unreasonable Success of Local Search: Geometric Optimization

TL;DR

This paper demonstrates that local search algorithms with sufficiently large neighborhoods serve as effective approximation schemes for several geometric problems in the Euclidean plane, under certain input assumptions.

Contribution

It proves that local search with neighborhoods of size proportional to $1/ ext{epsilon}^c$ provides approximation schemes for multiple geometric problems, including TSP, Steiner tree, facility location, and bicriteria k-median.

Findings

Local search yields approximation schemes for TSP with random inputs.

Local search provides approximation schemes for Steiner tree with random inputs.

Local search achieves approximation schemes for facility location and bicriteria k-median in worst-case inputs.

Abstract

What is the effectiveness of local search algorithms for geometric problems in the plane? We prove that local search with neighborhoods of magnitude $1/\epsilon^c$ is an approximation scheme for the following problems in the Euclidian plane: TSP with random inputs, Steiner tree with random inputs, facility location (with worst case inputs), and bicriteria $k$-median (also with worst case inputs). The randomness assumption is necessary for TSP.