A note on finding large transversals efficiently

TL;DR

This paper demonstrates that large transversals can be efficiently found in n x n arrays with bounded symbol frequency, providing a polynomial-time algorithm for constructing such transversals.

Contribution

It establishes a lower bound on the size of transversals in arrays with limited symbol repetitions and offers a deterministic polynomial-time algorithm to find them.

Findings

Transversals of size approximately 3/4 n exist in equi-n arrays.

A polynomial-time algorithm can find these large transversals.

The results apply to arrays with symbols appearing no more than βn times.

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 $\beta n$ times, the array contains a transversal of size $(1-\beta/4-o(1))n$. 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.