hw2_sol - 10-725 Optimization Spring 2008 Homework 2...

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Unformatted text preview: 10-725 Optimization, Spring 2008: Homework 2 Solutions Due: Wednesday, February 25, beginning of the class 1 Vertex Cover [Gaurav, 25 points] The goal of this problem is to illustrate the use of LPs for approximating NP-hard optimization problems. We will obtain an approximation to the vertex cover problem. Vertex Cover: Given an undirected graph G = ( V,E ) and a cost function on vertices c : V-→ Q + (positive rationals), find a set of vertices V ⊆ V of minimum cost, such that every edge has at least one endpoint incident at V . For example, consider the following graph. Assume all the vertices have equal weight. Then the problem reduces to picking the minimum number of vertices such that each edge is covered. For this graph, the minimum cost vertex cover is { B,E } . 1. [2 pts] Associate a binary (0/1) variable x i with each vertex i , which denotes whether the vertex has been picked in the vertex cover or not. Express the condition that each edge has at least one of its ends picked, as a linear constraint. F SOLUTION: For every edge, we want at least one end point to be part of the vertex cover. Therefore, we get the constraint x i + x j ≥ 1 ∀ ( i,j ) ∈ E 2. [2 pts] If all the vertices have equal weight, what is the objective function? F SOLUTION: We are trying to find the minimum vertex cover. Thus, the objective fuction is min x X i ∈ V x i 3. [2 pts] If the vertices have different weights, what is the new objective function? 1 F SOLUTION: min x X i ∈ V c i x i where c i is the weight of the vertex v i . 4. [3 pts] Write out an integer program (IP) that exactly solves the vertex cover problem. Hint: An integer program is one in which the objective function and constraints are linear, but there are some variables that are constrained to take only integer values . F SOLUTION: min X i ∈ V c i x i subject to x i + x j ≥ 1 ∀ ( i,j ) ∈ E x i ∈ { , 1 } The last constraint is what makes this optimization problem an integer program, instead of a linear program. 5. [3 pts] Integer Programming is an NP-hard problem. This means that there cannot exist any polynomial time algorithm for solving an integer program, unless P = NP . Therefore, we will try to get an approximate solution by obtaining an LP from the IP. This is known as an LP relaxation. To get an LP relaxation from an IP, we retain the objective function and constraints, except for the constraints that some variables have to take integer values. We replace these constraints by new constraints that say that the variables come from a continuous range. Obtain an LP relaxation of the above IP problem. F SOLUTION: We relax the last (integrality) constraint in the previous formulation to obtain the following linear program min X i ∈ V c i x i subject to x i + x j ≥ 1 ∀ ( i,j ) ∈ E x i ≥ x 1 ≤ 1 Since this is a minimization problem and all weights c i are positive, the first and second constraint above make the last constraint redundant....
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This note was uploaded on 05/25/2008 for the course MACHINE LE 10708 taught by Professor Carlosgustin during the Spring '07 term at Carnegie Mellon.

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hw2_sol - 10-725 Optimization Spring 2008 Homework 2...

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