Permanent of bipartite graphs in terms of determinants
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913840621420544 |
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| author | Chakrabartty, Surabhi Singh, Ranveer |
| author_facet | Chakrabartty, Surabhi Singh, Ranveer |
| contents | Computing the permanent of a $(0,1)$-matrix is a well-known $\#P$-complete problem. In this paper, we present an expression for the permanent of a bipartite graph in terms of the determinant of the graph and its subgraphs, obtained by successively removing rows and columns corresponding to vertices involved in vertex-disjoint $4k$-cycles. Our formula establishes a general relationship between the permanent and the determinant for any bipartite graph. Since computing the permanent of a biadjacency matrix is equivalent to counting the number of its perfect matchings, this approach also provides a more efficient method for counting perfect matchings in certain types of bipartite graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_08128 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Permanent of bipartite graphs in terms of determinants Chakrabartty, Surabhi Singh, Ranveer Discrete Mathematics 15A15, 05C31 G.2.2; G.2.1 Computing the permanent of a $(0,1)$-matrix is a well-known $\#P$-complete problem. In this paper, we present an expression for the permanent of a bipartite graph in terms of the determinant of the graph and its subgraphs, obtained by successively removing rows and columns corresponding to vertices involved in vertex-disjoint $4k$-cycles. Our formula establishes a general relationship between the permanent and the determinant for any bipartite graph. Since computing the permanent of a biadjacency matrix is equivalent to counting the number of its perfect matchings, this approach also provides a more efficient method for counting perfect matchings in certain types of bipartite graphs. |
| title | Permanent of bipartite graphs in terms of determinants |
| topic | Discrete Mathematics 15A15, 05C31 G.2.2; G.2.1 |
| url | https://arxiv.org/abs/2503.08128 |