Short lists for shortest descriptions in short time

TL;DR

This paper introduces an efficient algorithm that produces a polynomial-sized list of candidate descriptions for a binary string, approximating the shortest description despite Kolmogorov complexity's non-computability.

Contribution

It presents a novel polynomial-time algorithm for generating candidate descriptions, utilizing expander graphs and randomness dispersers, extending recent theoretical work.

Findings

Algorithm generates polynomial-sized list of descriptions

Uses expander graphs and dispersers for construction

Extends recent theoretical results in complexity theory

Abstract

Is it possible to find a shortest description for a binary string? The well-known answer is "no, Kolmogorov complexity is not computable." Faced with this barrier, one might instead seek a short list of candidates which includes a laconic description. Remarkably such approximations exist. This paper presents an efficient algorithm which generates a polynomial-size list containing an optimal description for a given input string. Along the way, we employ expander graphs and randomness dispersers to obtain an Explicit Online Matching Theorem for bipartite graphs and a refinement of Muchnik's Conditional Complexity Theorem. Our main result extends recent work by Bauwens, Mahklin, Vereschchagin, and Zimand.