  The linear complementarity problem (LCP) provides a unified approach to many
problems such as linear programs, convex quadratic programs, and bimatrix
games. The general LCP is known to be NP-hard, but there are some promising
results that suggest the possibility that the LCP with a P-matrix (PLCP) may be
polynomial-time solvable. However, no polynomial-time algorithm for the PLCP
has been found yet and the computational complexity of the PLCP remains open.
Simple principal pivoting (SPP) algorithms, also known as Bard-type algorithms,
are candidates for polynomial-time algorithms for the PLCP. In 1978, Stickney
and Watson interpreted SPP algorithms as a family of algorithms that seek the
sink of unique-sink orientations of $n$-cubes. They performed the enumeration
of the arising orientations of the $3$-cube, hereafter called
PLCP-orientations. In this paper, we present the enumeration of
PLCP-orientations of the $4$-cube.The enumeration is done via construction of
oriented matroids generalizing P-matrices and realizability classification of
oriented matroids.Some insights obtained in the computational experiments are
presented as well.
