@inproceedings{Mahajan1999ACA,
title={A Combinatorial Algorithm for Pfaffians},
author={Meena Mahajan and P. R. Subramanya and V. Vinay},
booktitle={Electronic Colloquium on Computational Complexity},
year={1999}
}

The Pfaffian of an oriented graph is closely linked to Perfect Matching. It is also naturally related to the determinant of an appropriately defined matrix. This relation between Pfaffian and determinant is usually exploited to give a fast algorithm for computing Pfaffians. We present the first completely combinatorial algorithm for computing the Pfaffian in polynomial time. Our algorithm works over arbitrary commutative rings. Over integers, we show that it can be computed in the complexity… CONTINUE READING