1,554,261 research outputs found
Exploiting -Closure in Kernelization Algorithms for Graph Problems
A graph is c-closed if every pair of vertices with at least c common
neighbors is adjacent. The c-closure of a graph G is the smallest number such
that G is c-closed. Fox et al. [ICALP '18] defined c-closure and investigated
it in the context of clique enumeration. We show that c-closure can be applied
in kernelization algorithms for several classic graph problems. We show that
Dominating Set admits a kernel of size k^O(c), that Induced Matching admits a
kernel with O(c^7*k^8) vertices, and that Irredundant Set admits a kernel with
O(c^(5/2)*k^3) vertices. Our kernelization exploits the fact that c-closed
graphs have polynomially-bounded Ramsey numbers, as we show
An efficient way to assemble finite element matrices in vector languages
Efficient Matlab codes in 2D and 3D have been proposed recently to assemble
finite element matrices. In this paper we present simple, compact and efficient
vectorized algorithms, which are variants of these codes, in arbitrary
dimension, without the use of any lower level language. They can be easily
implemented in many vector languages (e.g. Matlab, Octave, Python, Scilab, R,
Julia, C++ with STL,...). The principle of these techniques is general, we
present it for the assembly of several finite element matrices in arbitrary
dimension, in the P1 finite element case. We also provide an extension of the
algorithms to the case of a system of PDE's. Then we give an extension to
piecewise polynomials of higher order. We compare numerically the performance
of these algorithms in Matlab, Octave and Python, with that in FreeFEM++ and in
a compiled language such as C. Examples show that, unlike what is commonly
believed, the performance is not radically worse than that of C : in the
best/worst cases, selected vector languages are respectively 2.3/3.5 and
2.9/4.1 times slower than C in the scalar and vector cases. We also present
numerical results which illustrate the computational costs of these algorithms
compared to standard algorithms and to other recent ones
New recurrence relationships between orthogonal polynomials which lead to new Lanczos-type algorithms
Lanczos methods for solving Ax = b consist in constructing a sequence of vectors (Xk),k = 1,... such that rk = b-AXk= Pk(A)r0, where Pk is the orthogonal polynomial of degree at most k with respect to the linear functional c defined as c(εi) = (y, Air0). Let P(1)k be the regular monic polynomial of degree k belonging to the family of formal orthogonal polynomials (FOP) with respect to c(1) defined as c(1)(εi) = c(εi+1). All Lanczos-type algorithms are characterized by the choice of one or two recurrence relationships, one for Pk and one for P(1)k. We shall study some new recurrence relations involving these two polynomials and their possible combinations to obtain new Lanczos-type algorithms. We will show that some recurrence relations exist, but cannot be used to derive Lanczos-type algorithms, while others do not exist at all
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