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We give simplify the proofs of the 2 results in Marius Zimand's paper"Kolmogorov complexity version of Slepian-Wolf coding, proceedings of STOC2017, p22--32". The first is a universal polynomial time compression algorithm:on input $\varepsilon > 0$, a number $k$ and a string $x$ it computes inpolynomial time with probability $1-\varepsilon$ a program that outputs $x$ andhas length $k + O(\log^2 (|x|/\varepsilon))$, provided that there exists such aprogram of length at most $k$. The second result, is a distributed compressionalgorithm, in which several parties each send some string to a common receiver.Marius Zimand proved a variant of the Slepian-Wolf theorem using Kolmogorovcomplexity (in stead of Shannon entropy).  With our simpler proof we improve the parameters of Zimand's result.

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