1,511 research outputs found
Computing small pivot-minors.
A graph G contains a graph H as a pivot-minor if H can be obtained from G by applying a sequence of vertex deletions and edge pivots. Pivot-minors play an important role in the study of rank-width. However, so far, pivot-minors have only been studied from a structural perspective. We initiate a systematic study into their complexity aspects. We first prove that the PIVOT-MINOR problem, which asks if a given graph G contains a given graph H as a pivot-minor, is NP-complete. If H is not part of the input, we denote the problem by H-PIVOT-MINOR. We give a certifying polynomial-time algorithm for H -PIVOT-MINOR for every graph H with |V(H)|≤4|V(H)|≤4 except when H∈{K4,C3+P1,4P1}H∈{K4,C3+P1,4P1}, via a structural characterization of H-pivot-minor-free graphs in terms of a set FHFH of minimal forbidden induced subgraphs
On the complete pivoting conjecture for Hadamard matrices of small orders
In this paper we study explicitly the pivot structure of Hadamard matrices of small orders 16, 20 and 32. An algorithm computing the (n — j) x (n — j) minors of Hadamard matrices is presented and its implementation for n = 12 is described. Analytical tables summarizing the pivot patterns attained are given
Formalized linear algebra over Elementary Divisor Rings in Coq
This paper presents a Coq formalization of linear algebra over elementary
divisor rings, that is, rings where every matrix is equivalent to a matrix in
Smith normal form. The main results are the formalization that these rings
support essential operations of linear algebra, the classification theorem of
finitely presented modules over such rings and the uniqueness of the Smith
normal form up to multiplication by units. We present formally verified
algorithms computing this normal form on a variety of coefficient structures
including Euclidean domains and constructive principal ideal domains. We also
study different ways to extend B\'ezout domains in order to be able to compute
the Smith normal form of matrices. The extensions we consider are: adequacy
(i.e. the existence of a gdco operation), Krull dimension and
well-founded strict divisibility
Graphs of Small Rank-width are Pivot-minors of Graphs of Small Tree-width
We prove that every graph of rank-width is a pivot-minor of a graph of
tree-width at most . We also prove that graphs of rank-width at most 1,
equivalently distance-hereditary graphs, are exactly vertex-minors of trees,
and graphs of linear rank-width at most 1 are precisely vertex-minors of paths.
In addition, we show that bipartite graphs of rank-width at most 1 are exactly
pivot-minors of trees and bipartite graphs of linear rank-width at most 1 are
precisely pivot-minors of paths.Comment: 16 pages, 7 figure
Positroid Stratification of Orthogonal Grassmannian and ABJM Amplitudes
A novel understanding of scattering amplitudes in terms of on-shell diagrams
and positive Grassmannian has been recently established for four dimensional
Yang-Mills theories and three dimensional Chern-Simons theories of ABJM type.
We give a detailed construction of the positroid stratification of orthogonal
Grassmannian relevant for ABJM amplitudes. On-shell diagrams are classified by
pairing of external particles. We introduce a combinatorial aid called `OG
tableaux' and map each equivalence class of on-shell diagrams to a unique
tableau. The on-shell diagrams related to each other through BCFW bridging are
naturally grouped by the OG tableaux. Introducing suitably ordered BCFW bridges
and positive coordinates, we construct the complete coordinate charts to cover
the entire positive orthogonal Grassmannian for arbitrary number of external
particles. The graded counting of OG tableaux suggests that the positive
orthogonal Grassmannian constitutes a combinatorial polytope.Comment: 32 pages, 23 figures; v2. minor corrections; v3. several
clarifications and minor improvement
Fast and accurate con-eigenvalue algorithm for optimal rational approximations
The need to compute small con-eigenvalues and the associated con-eigenvectors
of positive-definite Cauchy matrices naturally arises when constructing
rational approximations with a (near) optimally small error.
Specifically, given a rational function with poles in the unit disk, a
rational approximation with poles in the unit disk may be obtained
from the th con-eigenvector of an Cauchy matrix, where the
associated con-eigenvalue gives the approximation error in the
norm. Unfortunately, standard algorithms do not accurately compute
small con-eigenvalues (and the associated con-eigenvectors) and, in particular,
yield few or no correct digits for con-eigenvalues smaller than the machine
roundoff. We develop a fast and accurate algorithm for computing
con-eigenvalues and con-eigenvectors of positive-definite Cauchy matrices,
yielding even the tiniest con-eigenvalues with high relative accuracy. The
algorithm computes the th con-eigenvalue in operations
and, since the con-eigenvalues of positive-definite Cauchy matrices decay
exponentially fast, we obtain (near) optimal rational approximations in
operations, where is the
approximation error in the norm. We derive error bounds
demonstrating high relative accuracy of the computed con-eigenvalues and the
high accuracy of the unit con-eigenvectors. We also provide examples of using
the algorithm to compute (near) optimal rational approximations of functions
with singularities and sharp transitions, where approximation errors close to
machine precision are obtained. Finally, we present numerical tests on random
(complex-valued) Cauchy matrices to show that the algorithm computes all the
con-eigenvalues and con-eigenvectors with nearly full precision
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