m j=1 in .NET Implement Data Matrix ECC200 in .NET m j=1

m j=1 use .net data matrix barcodes encoder tomake barcode data matrix with .net VB.NET wj Pr[clause Cj satis ed 2d Data Matrix barcode for .NET ] min(p, 1 p2 ). m j=1 wj min(p, 1 p2 ) OPT . We would like to extend result to all MAX SAT instances. To do this, we will use a this better bound on OPT than m wj . Assume that for every i the weight of the unit clause j=1 xi appearing in the instance is at least the weight of the unit clause xi ; this is without loss of generality since we could negate all occurrences of xi if the assumption is not true.

Let vi be the weight of the unit clause xi if it exists in the instance, and let vi be zero otherwise. m Lemma 5.6: OPT j=1 wj n vi .

i=1 Proof. For each i, the optimal solution can satisfy exactly one of xi and xi . Thus the weight of the optimal solution cannot include both the weight of the clause xi and the clause xi .

Since vi is the smaller of these two weights, the lemma follows. Theorem 5.7: We can obtain a randomized 1 ( 5 1)-approximation algorithm for MAX SAT.

2 Electronic web edition. Copyright 2010 by David P. Williamson and David B.

Shmoys. To be published by Cambridge University Press We can now extend the result..

Randomized rounding Proof. Let U be the set of indices of clauses of the instance excluding unit clauses of the form xi . As above, we assume without loss of generality that the weight of each clause xi is no greater than the weight of clause xi .

Thus j U wj = m wj n vi . Then set each xi to j=1 i=1 be true independently with probability p = 1 ( 5 1). Then 2 E[W ] = .

wj Pr[clause Cj satis ed .net framework gs1 datamatrix barcode ] wj Pr[clause Cj satis ed] wj (5.1).

p . m n wj vi p OPT, = p j=1 i=1 where (5.1) follows by T VS .NET barcode data matrix heorem 5.

5 and the fact that p = min(p, 1 p2 ). This algorithm can be derandomized using the method of conditional expectations..

Randomized rounding The algorithm of the pre visual .net gs1 datamatrix barcode vious section shows that biasing the probability with which we set xi true yields an improved approximation algorithm. However, we gave each variable the same bias.

In this section, we show that we can do still better by giving each variable its own bias. We do this by returning to the idea of randomized rounding, which we examined brie y in Section 1.7 in the context of the set cover problem.

Recall that in randomized rounding, we rst set up an integer programming formulation of the problem at hand in which there are 0-1 integer variables. In this case we will create an integer program with a 0-1 variable yi for each Boolean variable xi such that yi = 1 corresponds to xi set true. The integer program is relaxed to a linear program by replacing the constraints yi {0, 1} with 0 yi 1, and the linear programming relaxation is solved in polynomial time.

Recall that the central idea of randomized rounding is that the fractional value yi is interpreted as the probability that yi should be set to 1. In this case, we set each xi to true with probability yi independently. We now give an integer programming formulation of the MAX SAT problem.

In addition to the variables yi , we introduce a variable zj for each clause Cj that will be 1 if the clause is satis ed and 0 otherwise. For each clause Cj let Pj be the indices of the variables xi that occur positively in the clause, and let Nj be the indices of the variables xi that are negated in the clause. We denote the clause Cj by .

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