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2.4. THREE RELATIVELY ADVANCED TOPICS use .net framework code 39 full ascii drawer toproduce barcode 39 on .net Visal Basic .NET the foregoing n .net vs 2010 USS Code 39 otation P = S R , where S R = {x : y s.t.

(x, y) R}). We refer to such search problems by the name candid search problems. De nition 2.

30 (candid search problems): An algorithm A solves the candid search problem of the binary relation R if for every x S R (i.e., for every (x, y) R) it holds that (x, A(x)) R.

The time complexity of such an algorithm is de ned def as T A. S R (n) = maxx P {0,1}n {t A (x)}, where t A (x) is the running time of A(x) and T A S R (n) = 0 if .NET Code 39 P {0, 1}n = . Note that nothing is required when x S R : In particular, algorithm A may either output a wrong solution (although no solutions exist) or run for more than T A.

S R (. x. ) steps. The r st case can be essentially eliminated whenever R PC. Furthermore, for R PC, if we know the time complexity of algorithm A (e.

g., if we can compute T A. S R (n) in poly .NET Code 3/9 (n)-time), then we may modify A into an algorithm A that solves the (general) search problem of R (i.e.

, halts with a correct output on each input) in time T A (n) = T A. S R (n) + poly( n). However, we do not necessarily know the running time of an algorithm that we consider. Furthermore, as we shall see in Section 2.

4.2, the naive assumption by which we always know the running time of an algorithm that we design is not valid either. Decision problems with a promise.

In the context of decision problems, a promise problem is a relaxation in which one is only required to determine the status of instances that belong to a predetermined set, called the promise. The requirement of ef cient veri cation is adapted in an analogous manner. In view of the standard usage of the term, we refer to decision problems with a promise by the name promise problems.

Formally, promise problems refer to a three-way partition of the set of all strings into yes-instances, no-instances, and instances that violate the promise. Standard decision problems are obtained as a special case by insisting that all inputs are allowed (i.e.

, the promise is trivial). De nition 2.31 (promise problems): A promise problem consists of a pair of nonintersecting sets of strings, denoted (Syes , Sno ), and Syes Sno is called the promise.

The promise problem (Syes , Sno ) is solved by algorithm A if for every x Syes it holds that A(x) = 1 and for every x Sno it holds that A(x) = 0. The promise problem is in the promise problem extension of P if there exists a polynomial-time algorithm that solves it. The promise problem (Syes , Sno ) is in the promise problem extension of N P if there exists a polynomial p and a polynomial-time algorithm V such that the following two conditions hold: 1.

Completeness: For every x Syes , there exists y of length at most p(. x. ) such that V ( x, y) = 1. 2. Soundness: For every x Sno and every y, it holds that V (x, y) = 0.

We stress that for algorithms of polynomial-time complexity, it does not matter whether the time complexity is de ned only on inputs that satisfy the promise or on all strings (see footnote 26). Thus, the extended classes P and N P (like PF and PC) are invariant under this choice..

P, NP, AND NP-COMPLETENESS Reducibility am ong promise problems. The notion of a Cook-reduction extend naturally to promise problems, when postulating that a query that violates the promise (of the problem at the target of the reduction) may be answered arbitrarily.28 That is, the oracle machine should solve the original problem no matter how queries that violate the promise are answered.

The latter requirement is consistent with the conceptual meaning of reductions and promise problems. Recall that reductions capture procedures that make subroutine calls to an arbitrary procedure that solves the reduced problem. But, in the case of promise problems, such a solver may behave arbitrarily on instances that violate the promise.

We stress that the main property of a reduction is preserved (see Exercise 2.35): If the promise problem is Cook-reducible to a promise problem that is solvable in polynomial time, then is solvable in polynomial time. We warn that the extension of a complexity class to promise problems does not necessarily inherit the structural properties of the standard class.

For example, in contrast to Theorem 2.35, there exists promise problems in N P coN P such that every set in N P can be Cook-reduced to them: see Exercise 2.36.

Needless to say, N P = coN P does not seem to follow from Exercise 2.36. See further discussion at the end of 2.


. 2.4.1.

2. Applic bar code 39 for .NET ations The following discussion refers both to the decision and search versions of promise problems.

Recall that promise problems offer the most direct way of formulating natural computational problems (e.g., when referring to computational problems regarding graphs, the promise mandates that the input is a graph).

In addition to the foregoing application of promise problems, we mention their use in formulating the natural notion of a restriction of a computational problem to a subset of the instances. Speci cally, such a restriction means that the promise set of the restricted problem is a subset of the promise set of the unrestricted problem..

De nition 2.32 (restriction of computational problems): For any P P and binary relation R, we say that the search problem R with promise P is a restriction of the search problem R with promise P. We say that the promise problem (Syes , Sno ) is a restriction of the promise problem (Syes , Sno ) if both Syes Syes and Sno Sno hold.

For example, when we say that 3SAT is a restriction of SAT, we refer to the fact that the set of allowed instances is now restricted to 3CNF formulae (rather than to arbitrary CNF formulae). In both cases, the computational problem is to determine satis ability (or to nd a satisfying assignment), but the set of instances (i.e.

, the promise set) is further restricted in the case of 3SAT. The fact that a restricted problem is never harder than the original problem is captured by the fact that the restricted problem is reducible to the original one (via the identity mapping). Other uses and some reservations.

In addition to the two aforementioned generic uses of the notion of a promise problem, we mention that this notion provides adequate. It follows that Code 39 Full ASCII for .NET Karp-reductions among promise problems are not allowed to make queries that violate the promise. Speci cally, we say that the promise problem = ( yes , no ) is Karp-reducible to the promise problem = ( yes , no ) if there exists a polynomial-time mapping f such that for every x yes (resp.

, x no ) it holds that f (x) yes (resp., f (x) no )..

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