A note on finding large transversals efficiently

Abstract

In an $n \times n$ array filled with symbols, a transversal is a collection of entries with distinct rows, columns and symbols. In this note we show that if no symbol appears more than $βn$ times, the array contains a transversal of size $(1-β/4-o(1))n$. In particular, if the array is filled with $n$ symbols, each appearing $n$ times (an equi-$n$ square), we get transversals of size $(3/4-o(1))n$. Moreover, our proof gives a deterministic algorithm with polynomial running time, that finds these transversals.