4.1 Introduction 14.2 Linear Programming as a Lower Bounding Technique 14.2.1 Step 1: Reduce Problem to an Integer Program

Given an instance of Weighted Vertex Cover, we can encode it as an integer programming (IP) problem. Create a 0/1 indicator variable xv for each vertex v ∈ V : xv = 1 indicates v is in the cover; xv = 0 otherwise. We force {v : xv = 1} to be a cover by requiring that for each edge {u, v} ∈ E, xu = 1 or xv = 1 (or equivalently, xu + xv ≥ 1). Finally, since our objective is to find a minimum-cost vertex cover, we get the following IP: