  This paper begins with a class of convex quadratic programs (QPs) with
bounded variables solvable by the parametric principal pivoting algorithm with
$\mathcal{O}(n^3)$ strongly polynomial complexity, where $n$ is the number of
variables of the problem. Extension of the Hessian class is also discussed. Our
research is motivated by a recent reference [7] wherein the efficient solution
of a quadratic program with a tridiagonal Hessian matrix in the quadratic
objective is needed for the construction of a polynomial-time algorithm for
solving an associated sparse variable selection problem. With the tridiagonal
structure, the complexity of the QP algorithm reduces to $\mathcal{O}(n^2)$.
Our strongly polynomiality results extend previous works of some strongly
polynomially solvable linear complementarity problems with a P-matrix [9];
special cases of the extended results include weakly quasi-diagonally dominant
problems in addition to the tridiagonal ones.
