CS6320: Intro to Intractability, Approximation and Heuristic Algorithms Spring Lecture 20: April 6

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1 CS6320: Intro to Intractability, Approximation and Heuristic Algorithms Spring 2016 Lecture 20: April 6 Instructor: Prof. Ajay Gupta Scribe: Jason Pearson Many thanks to Teofilo Gonzalez for his lecture material. 1 Introduction to Vertex Cover Vertex cover is when given a graph determine if there is a subgraph that touches every edge such that the subgraph is minimized. This is the optimization version of the problem. The decision version would be is there a subgraph of size x where every edge in the graph is touched by nodes in the subgraph. Example: A university offers a number of classes. A student needs to meet requirements in order to get their degree. (i.e. taking and passing courses). Some requirements could be a math class, a class with a lab, a science class, an english class, and a computer class. The classes offered are as follows. Class Chemistry Physics Computer Programming Word Processing Creative Writing Calculus Requirements Met Science Science, Math, Class with Lab Computer Class, Class with Lab Computer Class, English Class English Class Math The goal of the student is to take as few classes as possible and cover all the requirements for the degree. Looking at the example the minimum cost vertex cover would be Physics and Word Processing. Vertex Cover Optimization Definition: Given an undirected graph G find the minimum number of verticies V V such that every edge is incident to at least one vertex in V. Then V would be the vertex cover of the graph. It is also known that V V creates an independent set since all edges are included in V. 1

2 2 Greedy Approximation Algorithm for Vertex Cover The greedy strategy that will be analyzed is simply to include nodes that have the highest degree. Remove that node and its respective edges and repeat until all edges have been removed. It will also be shown that the problem for this algorithm arises with its poor ability to handle ties. Graph G = (V,E) VC = empty set to start While an edge e E Pick a vertex u G of the largest degree V C = V C {v} delete all edges incident to V in G delete u from G End While Example 1: ˆf = 2 and f = 2 Example 2: ˆf = 3 and f = 4 2

3 Example 3: ˆf = 3 and f = 2. The optimal solution would be nodes 4 and 5 but the greedy algorithm would give the solution 1, 2, and 3. Example 4: ˆf = 8 and f = 4. In this example the same idea is used to generate the graph such that the optimal solution will be nodes 9-12 and the solution given by the greedy algorithm is nodes 1 through 8. 3

4 Example 5: ˆf = 20 and f = 8. The middle column is the optimal solution and the outer two columns are the greedy algorithm s solution. Now that a few examples have been given showing how the greedy solution is suboptimal a generic way to create a graph of this type can be created to show the bound on the greedy solution. For the graph the left side is 2 k nodes of degree 2 k this will be the optimal solution side. The right side consists of 2 k 1 nodes of degree 2 k, 2 k 1 nodes of degree 2 k 1,..., 2 k 1 nodes of degree 2 1, and then 2 k nodes of degree 1. Creating the graph in this way makes ˆf = k 2 k k and f = 2 k. We can use these values to get an approximation bound of ˆf/f = k 2k 1 +2 k = k/ We then 2 k solve for the number of nodes with respect to k so n = 2 k+1 + k 2 k 1 using this we can see that k = Ω (log (p)). This means that the approximation algorithm has a bound that scales with input size so it is not a good algorithm to use at all. 4

5 3 Using Integer Linear Programming as an Approximation Algorithm Given an undirected graph G = (V, E), find the least integer k such that the size of the vertex cover is K and every edge is adjacent to at least one vertex in V. Associate x i with vertex V i. x i = 1 if included in our vertex cover x i = 0 if not included in our vertex cover Rephrasing VC Find least integer k such that there is a vector X = [x 1, x 2, x 3,..., x n ] with x i = {0 1} n i=0 x i = k and for every edge e E is incident upon at least one vertex V i such that x i = 1 Minimize K subject to the constraints: x i + x j 1 (i, j) θ x i = {0 1} i 1 i n Example: Minimize x 1 + x 2 + x 3 + x 4 + x 5 Subject to: x 1 + x 3 1 x 1 + x 5 1 x 2 + x 4 1 x 3 + x 4 1 x 4 + x 5 1 For this example x 1 and x 4 are the nodes needed for the optimal solution. Integer linear programming cannot be used practically so there needs to be a way to convert a problem instance to a linear programming problem instance which are solvable in polynomial time. 5