Enumeration of PLCP-orientations of the 4-cube

Abstract

a b s t r a c tThe 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.

On the Holt-Klee Property for Oriented Matroid Programming

“…An application of this proposition concerns P-oriented matroid complementarity problems, introduced by Todd [22] and studied by Klaus and Miyata [15]. Here N is a rank n oriented matroid on the set [2n], satisfying the condition:…”

“…If we require N to satisfy property P , but do not require the extension N ∪ f to be realizable, we get what Klaus and Miyata [15] call a POMCP, where OMCP stands for oriented matroid complementarity problem. Our results imply that for such POMCP with the rank of N at most 4, the digraph satisfies the Holt-Klee property.…”

The Niceness of Unique Sink Orientations

“…First described by Stickney and Watson already in 1978 as abstract models for P-matrix linear complementarity problems (PLCPs) [29], USO were revived by Szabó and Welzl in 2001 [30]. Subsequently, their structural and algorithmic properties were studied extensively ( [27], [28], [23], [14], [7], [2], [17], [15], [20], [18]). In a nutshell, a USO is an orientation of the n-dimensional hypercube graph, with the property that there is a unique sink in every subgraph induced by a nonempty face.…”