Extending parameterized problem-tracing questions for Java with personalized guidance

2009 ◽  
Author(s):  
I-Han Hsiao ◽  
Sergey Sosnovsky ◽  
Peter Brusilovsky
Keyword(s):  
Parameterized Problem

Multiplicative Parameterization Above a Guarantee

2021 ◽  
Vol 13 (3) ◽  
pp. 1-16
Author(s):  
Fedor V. Fomin ◽  
Petr A. Golovach ◽  
Daniel Lokshtanov ◽  
Fahad Panolan ◽  
Saket Saurabh ◽  
...  
Keyword(s):  
Minimum Size ◽  
Input Graph ◽  
Fixed Parameter Tractable ◽  
Long Cycle ◽  
Fixed Parameter ◽  
Parameterized Problem ◽  
Fixed Constant ◽  
Fixed Parameter Algorithm ◽  
Np Hardness ◽  
Additive Form

Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance ( I,k ) of some (parameterized) problem π with a guarantee g(I) , decide whether I admits a solution of size at least (or at most) k + g(I) . Here, g(I) is usually a lower bound on the minimum size of a solution. Since its introduction in 1999 for M AX SAT and M AX C UT (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research. We highlight a multiplicative form of parameterization above (or, rather, times) a guarantee: Given an instance ( I,k ) of some (parameterized) problem π with a guarantee g(I) , decide whether I admits a solution of size at least (or at most) k · g(I) . In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, which is the most natural guarantee for this problem, and provide a fixed-parameter algorithm. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ε > 0, multiplicative parameterization above g(I) 1+ε of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of fixed-parameter algorithms as well as kernelization algorithms for additional problems parameterized multiplicatively above girth.

SOLUTION-BASED MESH QUALITY INDICATORS FOR TRIANGULAR AND TETRAHEDRAL MESHES

2000 ◽  
Vol 10 (03) ◽  
pp. 333-346 ◽  
Author(s):  
MARTIN BERZINS
Keyword(s):  
Quality Indicators ◽  
Quadratic Function ◽  
Local Solution ◽  
Quality Measure ◽  
Mesh Quality ◽  
Tetrahedral Meshes ◽  
Tetrahedral Elements ◽  
Parameterized Problem

A new mesh quality measure for triangular and tetrahedral meshes is presented. This mesh quality measure is based both on geometrical and solution information and is derived by considering the error when linear triangular and tetrahedral elements are used to approximate a quadratic function. The new measure is shown to be related to existing measures of mesh quality but with the advantage that local solution information in the form of scaled derivatives along edges is taken into account. This advantage is demonstrated by a comparison with a geometrical indicator on a parameterized problem.

scholarly journals On the complexity of Gödel's proof predicate

2010 ◽  
Vol 75 (1) ◽  
pp. 239-254 ◽  
Author(s):  
Yijia Chen ◽  
Jörg Flum
Keyword(s):  
Classical Problem ◽  
Order Logic ◽  
First Order Logic ◽  
Fixed Parameter Tractability ◽  
Fixed Parameter Tractable ◽  
First Order ◽  
Fixed Parameter ◽  
Parameterized Problem ◽  
Computable Bound ◽  
Increasing Functions

AbstractThe undecidability of first-order logic implies that there is no computable bound on the length of shortest proofs of valid sentences of first-order logic. Some valid sentences can only have quite long proofs. How hard is it to prove such “hard” valid sentences? The polynomial time tractability of this problem would imply the fixed-parameter tractability of the parameterized problem that, given a natural number n in unary as input and a first-order sentence φ as parameter, asks whether φ has a proof of length ≤ n. As the underlying classical problem has been considered by Gödel we denote this problem by p-Gödel. We show that p-Gödel is not fixed-parameter tractable if DTIME(hO(1)) ≠ NTIME(hO(1)) for all time constructible and increasing functions h. Moreover we analyze the complexity of the construction problem associated with p-Gödel.

Extending parameterized problem-tracing questions for Java with personalized guidance

2009 ◽  
Vol 41 (3) ◽  
pp. 392-392
Author(s):  
I-Han Hsiao ◽  
Sergey Sosnovsky ◽  
Peter Brusilovsky
Keyword(s):  
Parameterized Problem

scholarly journals MESHLESS GALERKIN METHOD BASED ON RBFS AND REPRODUCING KERNEL FOR QUASI-LINEAR PARABOLIC EQUATIONS WITH DIRICHLET BOUNDARY CONDITIONS

2021 ◽  
Vol 26 (2) ◽  
pp. 318-336
Author(s):  
Mehdi Mesrizadeh ◽  
Kamal Shanazari
Keyword(s):  
Hilbert Space ◽  
Boundary Conditions ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Solution Space ◽  
Dirichlet Boundary Conditions ◽  
Science And Engineering ◽  
Linear Parabolic Equations ◽  
Parameterized Problem ◽  
Space Method

The main aim of this paper is to present a hybrid scheme of both meshless Galerkin and reproducing kernel Hilbert space methods. The Galerkin meshless method is a powerful tool for solving a large class of multi-dimension problems. Reproducing kernel Hilbert space method is an extremely efficient approach to obtain an analytical solution for ordinary or partial differential equations appeared in vast areas of science and engineering. The error analysis and convergence show that the proposed mixed method is very efficient. Since the solution space spanned by radial basis functions do not directly satisfy essential boundary conditions, an auxiliary parameterized technique is employed. Theoretical studies indicate that this new method is very stable, though a parameterized problem is employed instead of the main problem.

scholarly journals Parameterized Extended Finite Element Method for high thermal gradients

2017 ◽  
Vol 5 (3) ◽  
pp. 329-336 ◽  
Author(s):  
Christian Zeller ◽  
Binu Surendran ◽  
Micheal F. Zaeh
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Element ◽  
Extended Finite Element Method ◽  
Thermal Gradients ◽  
Extended Finite Element ◽  
Time Step ◽  
High Gradients ◽  
Parameterized Problem ◽  
Element Method

Abstract The Finite Element Method results in inaccuracies for temperature changes at the boundary if the mesh is too coarse in comparison with the applied time step. Oscillations occur as the adjacent elements balance the excessive energy of the boundary element. An Extended Finite Element Method (XFEM) with extrinsic enrichment of the boundary element by a parameterized problem-specific ansatz function is presented. The method is able to represent high thermal gradients at the boundary with a coarse mesh as the enrichment function compensates for the excessive energy at the element affected by the temperature change. The parameterization covers the temporal change of the gradient and avoids the enrichment by further ansatz functions. The introduced parameterization variable is handed over to the system of equations as an additional degree of freedom. Analytical integration is used for the evaluation of the integrals in the weak formulation as the ansatz function depends non-linearly on the parameterization variable. Highlights Parameterized problem-specific ansatz functions. Avoidance of a fine mesh in the area of high gradients. Representation of high gradients with one additional DOF.

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