Optimal probabilistic polynomial time compression and the Slepian-Wolf theorem: tighter version and simple proofs

TL;DR

This paper simplifies the proofs of two key results in Kolmogorov complexity-based compression, improving parameters and providing a more accessible approach to universal polynomial-time compression and distributed Slepian-Wolf coding.

Contribution

The paper offers simplified proofs of existing results in Kolmogorov complexity-based compression, enhancing the parameters of Zimand's distributed Slepian-Wolf theorem.

Findings

Simplified proof of universal polynomial-time compression algorithm.

Improved parameters for distributed compression in Slepian-Wolf setting.

Enhanced accessibility of Kolmogorov complexity-based coding proofs.

Abstract

We give simplify the proofs of the 2 results in Marius Zimand's paper "Kolmogorov complexity version of Slepian-Wolf coding, proceedings of STOC 2017, p22--32". The first is a universal polynomial time compression algorithm: on input $\varepsilon > 0$, a number $k$ and a string $x$ it computes in polynomial time with probability $1-\varepsilon$ a program that outputs $x$ and has length $k + O(\log^2 (|x|/\varepsilon))$, provided that there exists such a program of length at most $k$. The second result, is a distributed compression algorithm, in which several parties each send some string to a common receiver. Marius Zimand proved a variant of the Slepian-Wolf theorem using Kolmogorov complexity (in stead of Shannon entropy). With our simpler proof we improve the parameters of Zimand's result.