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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Accesso online: | https://arxiv.org/abs/2601.18450 |
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| _version_ | 1866915756058345472 |
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| author | Abdurakhmanov, Jamolidin K. |
| author_facet | Abdurakhmanov, Jamolidin K. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18450 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the existence of heavy columns in binary matrices with distinct rows Abdurakhmanov, Jamolidin K. Discrete Mathematics 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. |
| title | On the existence of heavy columns in binary matrices with distinct rows |
| topic | Discrete Mathematics |
| url | https://arxiv.org/abs/2601.18450 |