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We describe two new 3/2-approximation algorithms and a new 2-approximation algorithm for the minimum weight edge cover problem in graphs. We show that one of the 3/2-approximation algorithms, the Dual cover algorithm, computes the lowest weight edge cover relative to previously known algorithms as well as the new algorithms reported here. The Dual cover algorithm can also be implemented to be faster than the other 3/2-approximation algorithms on serial computers. Many of these algorithms can be extended to solve the 6-Edge cover problem as well. We show the relation of these algorithms to the K-Nearest Neighbor graph construction in semi-supervised learning and other applications. 

We prove the equivalence of two different Hessian evaluation algorithms in AD. The first is the Edge Pushing algorithm of Gower and Mello, which may be viewed as a second order Reverse mode algorithm for computing the Hessian. In earlier work, we have derived the Edge Pushing algorithm by exploiting a Reverse mode invariant based on the concept of live variables in compiler theory. The second algorithm is based on eliminating vertices in a computational graph of the gradient, in which intermediate variables are successively eliminated from the graph, and the weights of the edges are updated suitably. %The algorithm can be performed on any order of eliminating vertices. We prove that if the vertices are eliminated in a reverse topological order while preserving symmetry in the computational graph of the gradient, then the Vertex Elimination algorithm and the Edge Pushing algorithm perform identical computations. In this sense, the two algorithms are equivalent. This insight that unifies two seemingly disparate approaches to Hessian computations could lead to improved algorithms and implementations for computing Hessians.