Using a Skewed Hamming Distance to Speed Up Deterministic Local Search

TL;DR

This paper introduces a modified deterministic local search algorithm for (d,k)-CSP problems that uses a skewed Hamming distance to significantly improve running time, approaching the efficiency of randomized algorithms.

Contribution

A simple modification to deterministic local search using a skewed Hamming distance, narrowing the gap with randomized algorithms for (d,k)-CSP problems.

Findings

Achieves a running time close to randomized algorithms for most d values.

Introduces a graph structure on colors to accelerate local search.

Provides a practical deterministic approach with improved efficiency.

Abstract

Schoening presents a simple randomized algorithm for (d,k)-CSP problems with running time (d(k-1)/k)^n poly(n). Here, d is the number of colors, k is the size of the constraints, and n is the number of variables. A derandomized version of this, given by Dantsin et al., achieves a running time of (dk/(k+1))^n poly(n), inferior to Schoening's. We come up with a simple modification of the deterministic algorithm, achieving a running time of (d(k-1)/k * k^d/(k^d-1))^n \poly(n). Though not completely eleminating the gap, this comes very close to the randomized bound for all but very small values of d. Our main idea is to define a graph structure on the set of d colors to speed up local search.