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Permanent of biadjacency of bipartite graphs is the number of perfect matchings. In the case of planar graphs we can obtain an orientation with sign changes and get away with computing the determinant ...

Suppose the sparsity pattern of $A \in \mathbb{R}^{N \times N}$ is a planar graph. Can I use this to bound the complexity of solving $$ Ax = b $$ ? In particular, I was hoping to use the planar ...