Let us remark here that a locally optimal independent set can be found in

polynomial time for the class of graphs for which a maximum independent set in [Gamma](v), [for every]v [element of] V, can be polynomially derived; this class includes, more particularly, the class of k-claw-free graphs for every fixed constant k.

101) Conversely, a problem is computationally in tractable if the (provably) optimal algorithm for solving the problem cannot solve all instances in

polynomial time.

Therefore, there is no algorithm can solve this problem in

polynomial time.

We present two

polynomial time algorithms for special cases.

The contributions that make up the main body of the text are devoted to an application of the effective Sato-Tate conjecture, Sato-Tate groups of some weight three motives, computing Hasse-Witt matrices of hyperelliptic cures in average

polynomial time, and a wide variety of other related subjects.

Multi-Constrained Any-path routing as demonstrated in [5] presents a

polynomial time K-approximation algorithm although fails to plan better approximation algorithm with strong hardness result on social networks.

2] -definable graph problems which cannot be solved on cliques in

polynomial time, unless EXP = NEXP.

The second half of the book develops algorithms for word search and conjugacy search problems, a decision algorithm for the work problem in free solvable groups, and a

polynomial time algorithm based on a straight line program.

Employing NP-hard problem such as SVP in stream cipher will strengthen the security of the generated keystream against cryptanalysis attacks, since there is no known method that can solve NP-hard problems in

polynomial time.

MC machines were proved to be powerful enough to solve NP problems in less than

polynomial time, see [1], [2], [3], [4], [5], [6] and [7]

We shall normally assume that any rule takes

polynomial time to apply.