On Sparse Matrix Chain Products

We investigate various combinatorial problems arising in the minimization of the number of arithmetic operations performed during the evaluation of matrix chain products. Sparsity as well as numerical equality of arbitrary entries of the matrix factors are taken into account by the Numerical Sparse Matrix Chain Product problem which turns out to be NP-complete. The computational complexity of the Structural Sparse Matrix Chain Product problem which assumes numerically distinct matrix entries remains open. Prior work on elimination techniques for the accumulation of Jacobian matrices in the context of algorithmic differentiation is built on.