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An introduction to the techniques and to linear programming: the set cover problem 17 in .NET Creation barcode data matrix in .NET An introduction to the techniques and to linear programming: the set cover problem 17

1.2 An introduction to the techniques and to linear programming: the set cover problem 17 generate, create data matrix 2d barcode none with .net projects Microsoft Official Website the problem Visual Studio .NET 2d Data Matrix barcode is an unweighted vertex cover problem. To see that the vertex cover problem is a special case of the set cover problem, for any instance of the vertex cover problem, create an instance of the set cover problem in which the ground set is the set of edges, and a subset Si of weight wi is created for each vertex i V containing the edges incident to i.

It is not di cult to see that for any vertex cover C, there is a set cover I = C of the same weight, and vice versa. A second thesis of this book is that linear programming plays a central role in the design and analysis of approximation algorithms. Many of the techniques introduced will use the theory of integer and linear programming in one way or another.

Here we will give a very brief introduction to the area in the context of the set cover problem; we give a slightly less brief introduction in Appendix A, and the notes at the end of this chapter provide suggestions of other, more in-depth, introductions to the topic. Each linear program or integer program is formulated in terms of some number of decision variables that represent some sort of decision that needs to be made. The variables are constrained by a number of linear inequalities and equalities called constraints.

Any assignment of real numbers to the variables such that all of the constraints are satis ed is called a feasible solution. In the case of the set cover problem, we need to decide which subsets Sj to use in the solution. We create a decision variable xj to represent this choice.

In this case we would like xj to be 1 if the set Sj is included in the solution, and 0 otherwise. Thus we introduce constraints xj 1 for all subsets Sj , and xj 0 for all subsets Sj . This is not su cient to guarantee that xj {0, 1}, so we will formulate the problem as an integer program to exclude fractional solutions (that is, non-integral solutions); in this case, we are also allowed to constrain the decision variables to be integers.

Requiring xj to be integer along with the constraints xj 0 and xj 1 is su cient to guarantee that xj {0, 1}. We also want to make sure that any feasible solution corresponds to a set cover, so we introduce additional constraints. In order to ensure that every element ei is covered, it must be the case that at least one of the subsets Sj containing ei is selected.

This will be the case if . j:ei Sj xj 1,. for each ei , i = 1, . . .

, n. In addition to the constraints, linear and integer programs are also de ned by a linear function of the decision variables called the objective function. The linear or integer program seeks to nd a feasible solution that either maximizes or minimizes this objective function.

Such a solution is called an optimal solution. The value of the objective function for a particular feasible solution is called the value of that solution. The value of the objective function for an optimal solution is called the value of the linear (or integer) program.

We say we solve the linear program if we nd an optimal solution. In the case of the set cover problem we want to nd a set cover of minimum weight. Given the decision variables xj and constraints described above, the weight of a set cover given the xj variables is m wj xj .

Thus the objective function of j=1 m the integer program is j=1 wj xj , and we wish to minimize this function. Integer and linear programs are usually written in a compact form stating rst the objective function, followed by the constraints. Given the discussion above, the problem of nding a Electronic web edition.

Copyright 2010 by David P. Williamson and David B. Shmoys.

To be published by Cambridge University Press.
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