Permanent of bipartite graphs in terms of determinants

TL;DR

The paper derives a general formula expressing the permanent of a bipartite graph in terms of determinants of reduced subgraphs obtained by removing the vertices of vertex-disjoint $4k$-cycles. The central result is that, for a bipartite graph with $n$ even and maximum number $m$ of disjoint $4k$-cycles, $\mathrm{per}(G) = (-1)^{n/2} \sum_{z=0}^{m} \frac{4^z}{z!} \sum_{T_z} \det(G \setminus T_z)$, where the inner sum ranges over ordered tuples of $z$ mutually vertex-disjoint $4k$-cycles and $G \setminus T_z$ is the graph obtained by removing the corresponding vertices. A key corollary is that if the graph is $4k$-cycle-free, then $\mathrm{per}(G) = (-1)^{n/2} \det(G)$, i.e., $|\mathrm{per}(G)|=|\det(G)|$. The authors discuss computational implications, particularly for counting perfect matchings and for identifying graph classes (e.g., bipartite cactus graphs) where the cycle enumeration is tractable, while acknowledging that enumerating all short cycles remains NP-hard in general. The work situates itself within the Pólya permanent problem and related Pfaffian techniques, and suggests future exploration of graph families for efficient application of the determinant-based formula.

Abstract

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.

Figures (1)

• Figure 1: Example of a bipartite graph