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Reducing Optimization Problems to Search Problems in .NET Development barcode 3 of 9 in .NET Reducing Optimization Problems to Search Problems .net vs 2010 bar code




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2.2.2. Reducing Optimization Problems to Search Problems generate, create none none in none projectsvb.net source code example barcode generate Many searc none for none h problems refer to a set of potential solutions, associated with each problem instance, such that different solutions are assigned different values (resp., costs ). In such a case, one may be interested in nding a solution that has value exceeding some threshold (resp.

, cost below some threshold). Alternatively, one may seek a solution of. .NET Framework SDK P, NP, AND NP-COMPLETENESS maximum va none none lue (resp., minimum cost). For simplicity, let us focus on the case of a value that we wish to maximize.

Still, there are two different objectives (i.e., exceeding a threshold and optimizing), giving rise to two different (auxiliary) search problems related to the same relation R.

Speci cally, for a binary relation R and a value function f : {0, 1} {0, 1} R, we consider two search problems. 1. Exceeding a threshold: Given a pair (x, v) the task is to nd y R(x) such that f (x, y) v, where R(x) = {y : (x, y) R}.

That is, we are actually referring to the search problem of the relation R f = {( x, v , y) : (x, y) R f (x, y) v},. (2.1). where x, v none none denotes a string that encodes the pair (x, v). 2. Maximization: Given x the task is to nd y R(x) such that f (x, y) = vx , where vx is the maximum value of f (x, y ) over all y R(x).

That is, we are actually referring to the search problem of the relation R f = {(x, y) R : f (x, y) = max { f (x, y )}}.. y R(x) def (2.2). Examples o f value functions include the size of a clique in a graph, the amount of ow in a network (with link capacities), etc. The task may be to nd a clique of size exceeding a given threshold in a given graph or to nd a maximum-size clique in a given graph. Note that, in these examples, the base search problem (i.

e., the relation R) is quite easy to solve, and the dif culty arises from the auxiliary condition on the value of a solution (presented in R f and R f ). Indeed, one may trivialize R (i.

e., let R(x) = {0, 1}poly(. x. ) for ever y x), and impose all necessary structure by the function f (see Exercise 2.8). We con ne ourselves to the case that f is polynomial-time computable, which in particular means that f (x, y) can be represented by a rational number of length polynomial in .

x. + . y. . We will show next that, in this case, the two aforementioned search problems (i.e.

, of R f and R f ) are computationally equivalent. Theorem 2.13: For any polynomial-time computable f : {0, 1} {0, 1} R and a polynomially bounded binary relation R, let R f and R f be as in Eq.

(2.1) and Eq. (2.

2), respectively. Then the search problems of R f and R f are computationally equivalent. Note that, for R PC and polynomial-time computable f , it holds that R f PC.

Combining Theorems 2.10 and 2.13, it follows that in this case both R f and R f are reducible to N P.

We note, however, that even in this case it does not necessarily hold that R f PC. See further discussion following the proof. Proof: The search problem of R f is reduced to the search problem of R f by nding an optimal solution (for the given instance) and comparing its value to the given threshold value.

That is, we construct an oracle machine that solves R f by making a single query to R f . Speci cally, on input (x, v), the machine issues the query x (to a solver for R f ), obtaining the optimal solution y (or an indication that R(x) = ), computes f (x, y), and returns y if f (x, y) v. Otherwise (i.

e., either y = or f (x, y) < v), the machine returns an indication that R f (x, v) = . Turning to the opposite direction, we reduce the search problem of R f to the search problem of R f by rst nding the optimal value vx = max y R(x) { f (x, y)}.

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