Enumeration of PLCP-orientations of the 4-cube

TL;DR

This paper enumerates all PLCP-orientations of the 4-cube, extending previous work on 3-cubes, using oriented matroids to explore the structure and complexity of these orientations.

Contribution

It provides the first enumeration of PLCP-orientations for the 4-cube, advancing understanding of the structure of P-matrix related orientations.

Findings

Enumeration of 4-cube PLCP-orientations completed

Insights into the structure of oriented matroids obtained

Computational experiments reveal new properties

Abstract

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.