Compressibility and Probabilistic Proofs

We consider several examples of probabilistic existence proofs using compressibility arguments, including some results that involve Lovász local lemma. 1 Probabilistic proofs: a toy example There are many well known probabilistic proofs that objects with some properties exist. Such a proof estimates the probability for a random object to violate the requirements and shows that it is small (or at least strictly less than 1). Let us look at a toy example. Consider a n × n Boolean matrix and its k × k minor (the intersection of k rows and k columns chosen arbitrarily). We say that the minor is monochromatic if all its elements are equal (either all zeros or all ones). Proposition. For large enough n and for k = O(log n), there exists a (n×n)matrix that does not contain a monochromatic (k × k)-minor. Proof. We repeat the same simple proof three times, in three different languages. (Probabilistic language) Let us choose matrix elements using independent tosses of a fair coin. For a given k colums and k rows, the probability of ∗LIRMM CNRS & University of Montpellier. On leave from IITP RAS, Moscow, Russia. E-mail address: [email protected]. Supported by ANR-15-CE40-001601 RaCAF grant. 1 ar X iv :1 70 3. 03 34 2v 1 [ cs .D M ] 9 M ar 2 01 7 getting a monochromatic minor at their intersection is 2−k 2+1. (Both zerominor and one-minor have probability 2−k 2 .) There are at most n choices for columns and the same number for rows, so by the union bound the probability of getting at least one monochromatic minor is bounded by n × n × 2−k2+1 = 2 logn−k2+1 = 2 logn−k)+1 and the last expression is less then 1 if, say, k = 3 log n and n is suffuciently large. (Combinatorial language) Let us count the number of bad matrices. For a given choice of columns and rows we have 2 possibilities for the minor and 2 2−k2 possibilities for the rest, and there is at most n choices for raws and columns, so the total number of matrices with monochromatic minor is n × n × 2× 2n2−k2 = 2n2+2k logn−k2+1 = 2n2+k(2 logn−k)+1, and this is less than 2 2 , the total number of Boolean (n× n)-matrices. (Compression language) To specify the matrix that has a monochromatic minor, it is enough to specify 2k numbers between 1 and n (rows and column numbers), the color of the monochromatic minor (0 or 1) and the remaining n − k bits in the matrix (their positions are already known). So we save k bits (compared to the straightforward list of all n bits) using 2k log n+ 1 bits instead (each number in the range 1 . . . n requires log n bits; to be exact, we may use dlog ne), so we can compress the matrix with a monochromatic minor if 2k log n+ 1 k, and not all matrices are compressible. Of course, these three arguments are the same: in the second one we multiply probabilities by 2 2 , and in the third one we take logarithms. However, the compression language provides some new viewpoint that may help our intuition. 2 A bit more interesting example In this example we want to put bits (zeros and ones) around the circle in a “essentially asymmetric” way: each rotation of the circle should change at least a fixed percentage of bits. More precisely, we are interested in the following statement: