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It s time to introduce some important concepts We know what a search problem is: its de ning characteristic is that any proposed solution can be quickly checked for correctness, in the sense that there is an ef cient checking algorithm C that takes as input the given instance I (the data specifying the problem to be solved), as well as the proposed solution S, and outputs true if and only if S really is a solution to instance I Moreover the running time of C(I, S) is bounded by a polynomial in |I|, the length of the instance We denote the class of all search problems by NP We ve seen many examples of NP search problems that are solvable in polynomial time In such cases, there is an algorithm that takes as input an instance I and has a running time polynomial in |I| If I has a solution, the algorithm returns such a solution; and if I has no solution, the algorithm correctly reports so The class of all search problems that can be solved in polynomial time is denoted P Hence, all the search problems on the right-hand side of the table are in P 239

Excel gives the copied sheet the same name as the original sheet, followed by a space and the number 2 or the next unused number in parentheses For example, the first copy of the sheet 2016 Planning is named 2016 Planning (2), the second copy is named 2016 Planning (3), and so on You can then rename the copied sheet by using this name to identify it for example:

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against unauthorized grant (or revoke) operations on the roles This is where you can see DBV protecting the security infrastructure that is, securing the components that enforce the security for the database and data itself In practice, realms are flexible and transparent The applications know nothing of the realms When implemented correctly, the standard security and application capabilities remain functioning This transparency is essential to an effective implementation so that well-behaved applications won t be negatively affected once the security capabilities are enabled

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set name of sheet "2016 Planning (2)" to "2018 Planning"

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Okay, P must stand for polynomial But why use the initials NP (the common chatroom abbreviation for no problem ) to describe the class of search problems, some of which are terribly hard NP stands for nondeterministic polynomial time, a term going back to the roots of complexity theory Intuitively, it means that a solution to any search problem can be found and veri ed in polynomial time by a special (and quite unrealistic) sort of algorithm, called a nondeterministic algorithm Such an algorithm has the power of guessing correctly at every step Incidentally, the original de nition of NP (and its most common usage to this day) was not as a class of search problems, but as a class of decision problems: algorithmic questions that can be answered by yes or no Example: Is there a truth assignment that satis es this Boolean formula But this too re ects a historical reality: At the time the theory of NPcompleteness was being developed, researchers in the theory of computation were interested in formal languages, a domain in which such decision problems are of central importance Are there search problems that cannot be solved in polynomial time In other words, is P = NP Most algorithms researchers think so It is hard to believe that exponential search can always be avoided, that a simple trick will crack all these hard problems, famously unsolved for decades and centuries And there is a good reason for mathematicians to believe that P = NP the task of nding a proof for a given mathematical assertion is a search problem and is therefore in NP (after all, when a formal proof of a mathematical statement is written out in excruciating detail, it can be checked mechanically, line by line, by an ef cient algorithm) So if P = NP, there would be an ef cient method to prove any theorem, thus eliminating the need for mathematicians! All in all, there are a variety of reasons why it is widely believed that P = NP However, proving this has turned out to be extremely dif cult, one of the deepest and most important unsolved puzzles of mathematics.

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