137,068 research outputs found
Perfect Elimination Orderings for Symmetric Matrices
We introduce a new class of structured symmetric matrices by extending the
notion of perfect elimination ordering from graphs to weighted graphs or
matrices. This offers a common framework capturing common vertex elimination
orderings of monotone families of chordal graphs, Robinsonian matrices and
ultrametrics. We give a structural characterization for matrices that admit
perfect elimination orderings in terms of forbidden substructures generalizing
chordless cycles in graphs.Comment: 16 pages, 3 figure
Vertex elimination orderings for hereditary graph classes
We provide a general method to prove the existence and compute efficiently
elimination orderings in graphs. Our method relies on several tools that were
known before, but that were not put together so far: the algorithm LexBFS due
to Rose, Tarjan and Lueker, one of its properties discovered by Berry and
Bordat, and a local decomposition property of graphs discovered by Maffray,
Trotignon and Vu\vskovi\'c. We use this method to prove the existence of
elimination orderings in several classes of graphs, and to compute them in
linear time. Some of the classes have already been studied, namely
even-hole-free graphs, square-theta-free Berge graphs, universally signable
graphs and wheel-free graphs. Some other classes are new. It turns out that all
the classes that we study in this paper can be defined by excluding some of the
so-called Truemper configurations. For several classes of graphs, we obtain
directly bounds on the chromatic number, or fast algorithms for the maximum
clique problem or the coloring problem
Generating facets for the cut polytope of a graph by triangular elimination
The cut polytope of a graph arises in many fields. Although much is known
about facets of the cut polytope of the complete graph, very little is known
for general graphs. The study of Bell inequalities in quantum information
science requires knowledge of the facets of the cut polytope of the complete
bipartite graph or, more generally, the complete k-partite graph. Lifting is a
central tool to prove certain inequalities are facet inducing for the cut
polytope. In this paper we introduce a lifting operation, named triangular
elimination, applicable to the cut polytope of a wide range of graphs.
Triangular elimination is a specific combination of zero-lifting and
Fourier-Motzkin elimination using the triangle inequality. We prove sufficient
conditions for the triangular elimination of facet inducing inequalities to be
facet inducing. The proof is based on a variation of the lifting lemma adapted
to general graphs. The result can be used to derive facet inducing inequalities
of the cut polytope of various graphs from those of the complete graph. We also
investigate the symmetry of facet inducing inequalities of the cut polytope of
the complete bipartite graph derived by triangular elimination.Comment: 19 pages, 1 figure; filled details of the proof of Theorem 4, made
many other small change
Recognizing sparse perfect elimination bipartite graphs
When applying Gaussian elimination to a sparse matrix, it is desirable to avoid turning zeros into non-zeros to preserve the sparsity. The class of perfect elimination bipartite graphs is closely related to square matrices that Gaussian elimination can be applied to without turning any zero into a non-zero. Existing literature on the recognition of this class and finding suitable pivots mainly focusses on time complexity. For matrices with m non-zero elements, the currently best known algorithm has a time complexity of . However, when viewed from a practical perspective, the space complexity also deserves attention: it may not be worthwhile to look for a suitable set of pivots for a sparse matrix if this requires space. We present two new algorithms for the recognition of sparse instances: one with a time complexity in space and one with a time complexity in space. Furthermore, if we allow only pivots on the diagonal, our second algorithm can easily be adapted to run in time
A note on perfect partial elimination
In Gaussian elimination it is often desirable to preserve existing zeros (sparsity). This is closely related to perfect elimination schemes on graphs. Such schemes can be found in polynomial time. Gaussian elimination uses a pivot for each column, so opportunities for preserving sparsity can be missed. In this paper we consider a more flexible process that selects a pivot for each nonzero to be eliminated and show that recognizing matrices that allow such perfect partial elimination schemes is NP-hard
On some simplicial elimination schemes for chordal graphs
We present here some results on particular elimination schemes for chordal
graphs, namely we show that for any chordal graph we can construct in linear
time a simplicial elimination scheme starting with a pending maximal clique
attached via a minimal separator maximal (resp. minimal) under inclusion among
all minimal separators
Bisimplicial edges in bipartite graphs
Bisimplicial edges in bipartite graphs are closely related to pivots in Gaussian elimination that avoid turning zeroes into non-zeroes. We present a new deterministic algorithm to nd such edges in bipartite graphs. The expected time complexity of our new algorithm is on random bipartite graphs in which each edge is present with a fixed probability p, a polynomial improvement over the fastest algorithm found in the existing literature
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