Increasing Kolmogorov Complexity

How much do we have to change a string to increase its Kolmogorov complexity. We show that we can increase the complexity of any non-random string of length n by flipping O( √ n) bits and some strings require Ω( √ n) bit flips. For a given m, we also give bounds for increasing the complexity of a string by flipping m bits. By using constructible expanding graphs we give an efficient algorithm that given any non-random string of length n will give a small list of strings of the same length, at least one of which will have higher Kolmogorov complexity. As an application, we show that BPP is contained in P relative to the set of Kolmogorov random strings. Allender, Buhrman, Koucký, van Melkbeek and Ronneberger [2] building on our techniques later improved this result to show that all of PSPACE reduces to P with an oracle for the random strings.