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Lecture 7 1 Linear Programming Relaxations

Stanford University CS261: Optimization Handout 7. Luca Trevisan January 25, 2011. Lecture 7. In which we show how to use Linear Programming to approximate the vertex cover problem. 1 Linear Programming Relaxations An integer Linear program (abbreviated ILP) is a Linear program (abbreviated LP). with the additional constraints that the variables must take integer values. For ex- ample, the following is an ILP: maximize x1 x2 + 2x3. subject to x1 x2 1. x2 + x3 2 (1). x1 N. x2 N. x3 N. Where N = {0, 1, 2, ..} is the set of natural numbers. The advantage of ILPs is that they are a very expressive language to formulate opti- mization problems, and they can capture in a natural and direct way a large number of combinatorial optimization problems.

The vertex cover approximation algorithm based on linear programming is very ele-gant and simple, but it requires the solution of a linear program. Our previous vertex cover approximation algorithm, instead, had a very fast linear-time implementation.

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