253 research outputs found
Enumeration of PLCP-orientations of the 4-cube
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 -cubes. They performed the enumeration
of the arising orientations of the -cube, hereafter called
PLCP-orientations. In this paper, we present the enumeration of
PLCP-orientations of the -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
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
On the number of simple arrangements of five double pseudolines
We describe an incremental algorithm to enumerate the isomorphism classes of
double pseudoline arrangements. The correction of our algorithm is based on the
connectedness under mutations of the spaces of one-extensions of double
pseudoline arrangements, proved in this paper. Counting results derived from an
implementation of our algorithm are also reported.Comment: 24 pages, 16 figures, 6 table
Computing the Face Lattice of a Polytope from its Vertex-Facet Incidences
We give an algorithm that constructs the Hasse diagram of the face lattice of
a convex polytope P from its vertex-facet incidences in time O(min{n,m}*a*f),
where n is the number of vertices, m is the number of facets, a is the number
of vertex-facet incidences, and f is the total number of faces of P. This
improves results of Fukuda and Rosta (1994), who described an algorithm for
enumerating all faces of a d-polytope in O(min{n,m}*d*f^2) steps. For simple or
simplicial d-polytopes our algorithm can be specialized to run in time
O(d*a*f). Furthermore, applications of the algorithm to other atomic lattices
are discussed, e.g., to face lattices of oriented matroids.Comment: 14 pages; to appear in: Comput. Geom.; the new version contains some
minor extensions and corrections as well as a more detailed treatment of
oriented matroid
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