This paper addresses a combinatorial optimization problem (COP), namely a variant of the (standard) matrix chain product (MCP) problem where the matrices are square and either dense (i.e. full) or lower/upper triangular. Given a matrix chain of length n, we first present a dynamic programming algorithm (DPA) adapted from the well known standard algorithm and having the same O(n
3) complexity. We then design and analyse two optimal O(n) greedy algorithms leading in general to different optimal solutions i.e. chain parenthesizations. Afterwards, we establish a comparison between these two algorithms based on the parallel computing of the matrix chain product through intra and inter-subchains coarse grain parallelism. Finally, an experimental study illustrates the theoretical parallel performances of the designed algorithms.