Search-To-Decision Reductions for Kolmogorov Complexity

A long-standing open problem dating back to the 1960s is whether there exists a search-to-decision reduction for the time-bounded Kolmogorov complexity problem – that is, the problem of determining whether the length of the shortest time-t program generating a given string x is at most s. In this work, we consider the more “robust” version of the time-bounded Kolmogorov complexity problem, referred to as the GapMINKT problem, where given a size bound s and a running time bound t, the goal is to determine whether there exists a poly(t, |x|)-time program of length s + O(log |x|) that generates x. We present the first non-trivial search-to-decision reduction R for the GapMINKT problem; R has a running-time bound of 2^ϵn for any ϵ > 0 and additionally only queries its oracle on “thresholds” s of size s + O(log |x|). As such, we get that any algorithm with running-time (resp. circuit size) 2^αspoly(|x|, t, s) for solving GapMINKT (given an instance (x, t, s), yields an algorithm for finding a witness with running-time (resp. circuit size) 2^(α+ϵ)spoly(|x|, t, s). Our second result is a polynomial-time search-to-decision reduction for the time-bounded Kolmogorov complexity problem in the average-case regime. Such a reduction was recently shown by Liu and Pass (FOCS’20), heavily relying on cryptographic techniques. Our reduction is more direct and additionally has the advantage of being length-preserving, and as such also applies in the exponential time/size regime. A central component in both of these results is the use of Kolmogorov and Levin’s Symmetry of Information Theorem.