A theoretical breakthrough for the problem of computing the cardinality of groups of the type E ( \ mathbb { F } _ q ) was achieved by Ren?Schoof, who, in 1985, published the first deterministic polynomial time algorithm.
42.
Suppose the decisional Diffie Hellman assumption holds for \ mathbb F _ p, "'Naor and Reingold "'show that for every probabilistic polynomial time algorithm \ mathcal { A } and sufficiently large " n"
43.
In orthogonally convex polygons, the number of rectangles in a minimum covering is equal to the number of blocks in an anti rectangle, and this fact can be used to build a polynomial time algorithm for finding a minimum covering by rectangles.
44.
At the same time, the graph genus problem is fixed-parameter tractable, i . e ., polynomial time algorithms are known to check whether a graph can be embedded into a surface of a given fixed genus as well as to find the embedding.
45.
Jaruphongsa summarized that " Lee et al . ( 2001 ) generalize the classical ( single echelon ) dynamic lot-sizing model to consider demand time windows, and they provide polynomial time algorithms for two cases where backorders are allowed and where they are not ."
46.
However, this is not a polynomial time algorithm because the number of digits in the solution may be as large as " " n ", far larger than a polynomial in the number of digits in the input value " n ".
47.
It means that if finding collisions would be feasible in polynomial time by algorithm A, we could find and use polynomial time algorithm R ( reduction algorithm ) that would use algorithm A to solve problem P, which is widely supposed to be unsolvable in polynomial time.
48.
This characterization can be used to prove the existence of a polynomial time algorithm that tests for the existence of a planar cover, by searching for each of the forbidden minors and returning that a planar cover exists only if this search fails to find any of them.
49.
The forbidden minor characterization of linkless graphs leads to a polynomial time algorithm for their recognition, but not for actually constructing an embedding . described a linear time algorithm that tests whether a graph is linklessly embeddable and, if so, constructs a flat embedding of the graph.
50.
Polynomial time algorithms are also known for finding a coloring matching this bound, and for finding optimal colorings of special classes of graphs, but the more general problem of deciding whether an arbitrary graph has an equitable coloring with a given number of colors is NP-complete.
