Exact Permanent Computation for Block-Structured Matrices

The matrix permanent is #P-complete in the general case, with no known polynomial-time algorithm. Classical exact methods—Ryser’s inclusion-exclusion formula and Glynn’s ±1 enumeration—require O(2N · N) operations, making unstructured instances beyond N ≈ 40 intractable on commodity hardware. We show that matrices with block-diagonal structure, identified via connected-component decomposition, admit exact permanent computation through multiplicative factorization: perm(A) = ∏k perm(Akk). Each block is solved independently using a shared-nothing parallel implementation of Glynn’s formula traversing the Gray-code enumeration with O(N) per-step rank-one updates on an aligned L1-resident array. We validate the implementation against Ryser’s formula at N = 24, obtaining exact agreement (perm(B1) = 2,737,155,960,793). We then demonstrate exact permanent computation for matrices of size N = 24 to 128 on ≈0.02 sec. Min.