Linear complementarity problems solvable by a polynomially bounded pivoting algorithm

A sufficient condition is given under which the parametric principal pivoting algorithm will compute the unique solution to a linear complementarity problem defined by an n by n P-matrix in no more than n pivots. The condition is then shown to be satisfied by a P-matrix which has a hidden Z transpose and thus in particular, by an H-matrix with positive diagonals as well as by a strictly diagonally dominant matrix. The same condition is also shown to be sufficient for Lemke’s almost complementary algorithm to compute a solution to a linear complementarity problem defined by an n by n nondegenerate matrix in at most n+1 pivots. Finally, a polynomial testing procedure for the condition is described.

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