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

Jamolidin K. Abdurakhmanov

TL;DR

This work investigates when a binary matrix with distinct rows must contain a heavy column, defined as having at least $\lceil m/2\rceil$ ones. It introduces two recursive algorithms, $A_1$ and $A_2$, that inspect submatrices formed by row-filtering and column-deletion, with $A_2$ featuring an early termination that immediately identifies a heavy column if exactly one row has a zero in some column. The authors prove that $A_1$ returning True guarantees a heavy column (Theorem 1), and, under stronger conditions (distinct columns and no all-zero columns), $A_2$ returning True also guarantees a heavy column (Theorem 2). The proofs hinge on constructing paths through the recursion trees via conjugate and unpaired rows, a novel technique in binary-matrix theory, with implications for voting theory, coding theory, cryptography, data mining, and combinatorial optimization. These results deepen our understanding of structural guarantees for column weights in binary matrices and suggest several avenues for algorithmic development and generalization.

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.

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

TL;DR

This work investigates when a binary matrix with distinct rows must contain a heavy column, defined as having at least ones. It introduces two recursive algorithms, and , that inspect submatrices formed by row-filtering and column-deletion, with featuring an early termination that immediately identifies a heavy column if exactly one row has a zero in some column. The authors prove that returning True guarantees a heavy column (Theorem 1), and, under stronger conditions (distinct columns and no all-zero columns), returning True also guarantees a heavy column (Theorem 2). The proofs hinge on constructing paths through the recursion trees via conjugate and unpaired rows, a novel technique in binary-matrix theory, with implications for voting theory, coding theory, cryptography, data mining, and combinatorial optimization. These results deepen our understanding of structural guarantees for column weights in binary matrices and suggest several avenues for algorithmic development and generalization.

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.
Paper Structure (17 sections, 3 theorems, 3 equations)

This paper contains 17 sections, 3 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Related Work and Novelty
  3. Classical Results on Binary Matrices
  4. Tournament Matrices and Score Sequences
  5. Novelty of Our Results
  6. Applications
  7. Voting Theory and Social Choice
  8. Coding Theory
  9. Cryptography
  10. Data Mining
  11. Set Cover and Combinatorial Optimization
  12. Preliminaries
  13. Algorithm $A_1$ and Theorem 1
  14. Algorithm $A_2$ and Theorem 2
  15. Conclusion
  16. ...and 2 more sections

Key Result

Theorem 1

Let $M$ be a binary matrix of size $m \times n$$(m \geq 1, n \geq 1)$ with distinct rows. If $A_1(M, m, n) = \mathop{\mathrm{True}}\nolimits$, then $M$ contains at least one heavy column.

Theorems & Definitions (16)

  • Definition 1: Heavy column
  • Definition 2: Row indexing
  • Definition 3: Number of rows notation
  • Definition 4: Submatrix sets $S_k^0$, $S_k^1$, and $S_k$
  • Definition 5: Conjugate rows
  • Definition 6: Unpaired row
  • Definition 7: Matrix reduction
  • Definition 8: Sequential reduction
  • Definition 9: Set of consistent rows
  • Theorem 1
  • ...and 6 more