Algorithm Design

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Chapter 11 Approximation Algorithms
(11.18) The set I s of short paths selected by the approximation algorithm,
and the lengths ~, satisfy the relation ~e ~e /32 for all i E I*-I.
Summing over all paths in I*-I, we get
iaI*--I
On the other hand, each edge is used by at most two paths in the solution I*,
so we have
~ g(P~) _< ~ 2~e-
~I*--I
Combining these bounds with (11.18) we get
/321I* I _ b. By this notation,
we mean that each coordinate of the vector Ax should be greater than or equal
to the corresponding coordinate of the vector b. Such systems of inequalities
define regions in space. For example, suppose x = (xl, x2) is a two-dimensional
vector, and we have the four inequalities
x~_> 0,x 2 >_0
x 1 + 2x 2 >_ 6
2X 1 + x 2 _> 6
Then the set of solutions is the region in the plane shown in Figure 11.10.
Given a region defined by Ax > b, linear programming seeks to minimize
a linear combination of the coordinates of x, over all x belonging to the region.
Such a linear combination can be written ctx, where c is a vector of coefficients,
and ctx denotes the inner product of two vectors. Thus our standard form for
Linear Programming, as an optimization problem, will be the following.
Given an m x n matrix A, and vectors b ~ R rn and c ~ R n, find a vector
x ~ R n to solve the following optimization problem:
min(ctx such that x > 0; Ax > b).
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