On the existence of heavy columns in binary matrices with distinct rows

TL;DR

This paper explores the conditions under which binary matrices with distinct rows contain heavy columns, introducing recursive algorithms that identify such columns based on matrix properties and novel proof techniques.

Contribution

The paper presents two recursive algorithms, A1 and A2, that determine the existence of heavy columns in binary matrices with distinct rows, with A2 featuring an innovative early termination condition.

Findings

Algorithm A1 confirms the presence of heavy columns when it returns True.

Algorithm A2 guarantees heavy columns under specific conditions, including no all-zero columns.

The proofs utilize a novel argument involving unpaired rows and recursive matrix analysis.

Abstract

We investigate the existence of heavy columns in binary matrices with distinct rows. A column of an m x n binary matrix is called heavy if the number of ones in it is at least m/2. We introduce two recursive algorithms, A1 and A2, that examine properties of subma trices obtained by row filtering and column deletion. We prove that if algorithm A1 returns True for a binary matrix with distinct rows, then the matrix contains at least one heavy column (Theorem 1). Further more, we prove that if algorithm A2 returns True for a binary matrix with distinct rows, distinct columns, and no all-zero columns, then the matrix also contains at least one heavy column (Theorem 2). The key innovation in A2 is an early termination condition: if exactly one row has a zero in some column, that column is immediately identified as heavy. The proofs employ a novel argument based on the existence of unpaired rows with respect to specific columns, combined with careful analysis of the recursive structure of the algorithms.