We generalize new criss-cross type algorithms for linear complementarity problems (LCPs) given with sufficient matrices. Most LCP solvers require a priori information about the input matrix. The sufficiency of a matrix is hard to be checked (no polynomial time method is known). Our algorithm is similar to Zhang's linear programming and Akkeleş, Balogh and Illés criss-cross type algorithm for LCP-QP problems. We modify our basic algorithm in such a way that it can start with any matrix M, without having any information about the properties of the matrix (sufficiency, bisymmetry, positive definiteness, etc.) in advance. Even in this case, our algorithm terminates with one of the following cases in a finite number of steps: it solves the LCP problem, it solves its dual problem or it gives a certificate that the input matrix is not sufficient, thus cycling can occur. Although our algorithm is more general that that of Akkeleş, Balogh and Illés's, the finiteness proof has been simplified. Furthermore, the finiteness proof of our algorithm gives a new, constructive proof to Fukuda and Terlaky's LCP duality theorem as well.
