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Chapter 13 Linear programming in Low Dimensions

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At the sight of the still intact city, he remembered his great international precursors and set the whole place on fire with his artillery in order that those who came after him might work off their excess energies in rebuilding. – The tin drum, Gunter Grass. In this chapter, we shortly describe (and analyze) a simple randomized algorithm for linear programming in low dimensions. Next, we show how to extend this algorithm to solve linear programming with violations. Finally, we will show how one can efficiently approximate the number constraints one needs to violate to make a linear program feasible. This serves as a fruitful ground to demonstrate some the techniques we visited already. Our discussion is going to be somewhat intuitive. We will fill in the details, and prove correctness of our algorithms formally in the next chapter. 13.1 Linear Programming Assume we are given a set of n linear inequalities defined of the form a1x1 + · · · + adxd ≤ b, where a1,..., ad, b are constants, and x1,..., xd are the variables. In the linear programming (LP) problem, one has to find a feasible solution; that is, a point (x1,..., xd) for which all the linear inequalities hold. In fact, usually we would like to find a feasible point that maximizes a linear expression (referred to as the target function of the LP) of the form c1x1 + · · · + cdxd, where c1,..., cd are prespecified constants. The set of points complying with a linear inequality a1x1 + · · · + adxd ≤ b, is just a halfspace of IR 3y + 2x ≤ 6 d, having the hyperplane a1x1 + · · · + adxd = b as a boundary, see figure on the right. As such, the feasible region of the LP is the intersection of n halfspaces; that is, it is a polyhedron. The linear target function is no more than specifying a direction, such that we need to find the point inside the polyhedron which is extreme i

Year: 2010
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