In parameterized complexity each problem instance comes with a parameter k and the parameterized problem is said to admit a polynomial kernel if there are polynomial time preprocessing rules that reduce the input instance down to an instance with size polynomial in k. Many problems have been shown to admit polynomial kernels, but it is only recently that a framework for showing the non-existenc...

A Class of new methods based on a septic non-polynomial spline function for the numerical solution of problems in calculus of variations is presented. The local truncation errors and the methods of order 2th, 4th, 6th, 8th, 10th, and 12th, are obtained. The inverse of some band matrixes are obtained which are required in proving the convergence analysis of the presented method. Convergence anal...

We present a sixth-order non-polynomial spline method for the solutions of two-point nonlinear boundary value problem u(4)+f(x,u)=0, u(a)=α1, u''(a)= α2, u(b)= β1,u''(b)= β2, in off step points. Numerical method of sixth-order with end conditions of the order 6 is derived. The convergence analysis of the method has been discussed. Numerical examples are presented to illustrate the applications ...

Numerical polynomial algebra emerges as a growing field of study in recent years with a broad spectrum of applications and many robust algorithms. Among the challenges in solving polynomial algebra problems with floating-point arithmetic, difficulties frequently arise in regularizing ill-posedness and handling large matrices. We elaborate regularization principles for reformulating the illposed...

This is the final article in a series of four articles. Richard Karp has proven that a deterministic polynomial time solution to K-SAT will result in a deterministic polynomial time solution to all NP-Complete problems. However, it is demonstrated that a deterministic polynomial time solution to any NP-Complete problem does not necessarily produce a deterministic polynomial time solution to all...

This is the final article in a series of four articles. Richard Karp has proven that a deterministic polynomial time solution to K-SAT will result in a deterministic polynomial time solution to all NP-Complete problems. However, it is demonstrated that a deterministic polynomial time solution to any NP-Complete problem does not necessarily produce a deterministic polynomial time solution to all...

This is the final article in a series of four articles. Richard Karp has proven that a deterministic polynomial time solution to K-SAT will result in a deterministic polynomial time solution to all NP-Complete problems. However, it is demonstrated that a deterministic polynomial time solution to any NP-Complete problem does not necessarily produce a deterministic polynomial time solution to all...

We extend the concept of polynomial time approximation algorithms to apply to problems for hierarchically speciied graphs, many of which are PSPACE-complete. Assuming P 6 = PSPACE, the existence or nonexistence of such eecient approximation algorithms is characterized, for several standard graph theoretic and combinatorial problems. We present polynomial time approximation algorithms for severa...

This is the final article in a series of four articles. Richard Karp has proven that a deterministic polynomial time solution to K-SAT will result in a deterministic polynomial time solution to all NP-Complete problems. However, it is demonstrated that a deterministic polynomial time solution to any NP-Complete problem does not necessarily produce a deterministic polynomial time solution to all...

We prove a number of results around kernelization of problems parameterized by the size of a given vertex cover of the input graph. We provide three sets of simple general conditions characterizing problems admitting kernels of polynomial size. Our characterizations not only give generic explanations for the existence of many known polynomial kernels for problems like q-Coloring, Odd Cycle Tran...