# On the Determinant and its Derivatives of the Rank-one Corrected Generator of a Markov Chain on a Graph

## TL;DR

This paper introduces an efficient algorithm to compute the determinant and its derivatives of a rank-one corrected generator matrix in a Markov chain, facilitating solutions to the Hamiltonian cycle problem.

## Contribution

It presents a novel method leveraging the matrix structure to compute all cofactors from a single LU decomposition, reducing computational complexity.

## Key findings

- Efficient computation of derivatives for the generator matrix.
- Reduction of cofactor calculations to a single LU decomposition.
- Application to solving the Hamiltonian cycle problem.

## Abstract

We present an algorithm to find the determinant and its first and second derivatives of a rank-one corrected generator matrix of a doubly stochastic Markov chain. The motivation arises from the fact that the global minimiser of this determinant solves the Hamiltonian cycle problem. It is essential for algorithms that find global minimisers to evaluate both first and second derivatives at every iteration. Potentially the computation of these derivatives could require an overwhelming amount of work since for the Hessian $N^2$ cofactors are required. We show how the doubly stochastic structure and the properties of the objective may be exploited to calculate all cofactors from a single LU decomposition.