A Double-Exponential Lower Bound for the Distinct Vectors Problem

In the (binary) DISTINCT VECTORS problem we are given a binary matrix A with pairwise different rows and want to select at most k columns such that, restricting the matrix to these columns, all rows are still pairwise different. A result by Froese et al. [JCSS] implies a 22O(k)poly(jAj)-time brute-force algorithm for DISTINCT VECTORS. We show that this running time bound is essentially optimal by showing that there is a constant c such that the existence of an algorithm solving DISTINCT VECTORS with running time 2O(2ck)poly(jAj) would contradict the Exponential Time Hypothesis.