(5/5)

# Given a partition of V into sets, the basic step of the algorithm, called flip

INSTRUCTIONS TO CANDIDATES

1. Vazirani, Exercise 2.2

2.2 Consider the following algorithm for the maximum cut problem, based on the technique of local search. Given a partition of V into sets, the basic step of the algorithm, called flip, is that of moving a vertex from one side of the partition to the other. The following algorithm finds a locally optimal solution under the flip operation, i.e., a solution which cannot be improved by a single flip.

The algorithm starts with an arbitrary partition of V. While there is a vertex such that flipping it increases the size of the cut, the algorithm flips such a vertex. (Observe that a vertex qualifies for a flip if it has more neigh- bors in its own partition than in the other side.) The algorithm terminates when no vertex qualifies for a flip. Show that this algorithm terminates in polynomial time, and achieves an approximation guarantee of 1/2.

2. Vazirani, Exercise 3.1

3.1 The hardness of the Steiner tree problem lies in determining the optimal subset of Steiner vertices that need to be included in the tree. Show this by proving that if this set is provided, then the optimal Steiner tree can be computed in polynomial time.

Hint: Find an MST on the union of this set and the set of required vertices. 3. Vazirani, Exercise 3.3

3.3 Give an approximation factor preserving reduction from the set cover problem to the following problem, thereby showing that it is unlikely to have a better approximation guarantee than O(logn).

Problem 3.14 (Directed Steiner tree) G = (V,E) is a directed graph with nonnegative edge costs. The vertex set V is partitioned into two sets, required and Steiner. One of the required vertices, r, is special. The problem is to find a minimum cost tree in G rooted into r that contains all the required vertices and any subset of the Steiner vertices.

Hint: Construct a three layer graph: layer 1 contains a required vertex corresponding to each element, layer 2 contains a Steiner vertex corresponding to each set, and layer 3 contains r.

4. Vazirani, Exercise 4.1.

4.1 Show that Algorithm 4.3 can be used as a subroutine for finding a k-cut within a factor of 2-2/k of the minimum k-cut. How many subroutine calls are needed?

(5/5)

## Related Questions

##### . Introgramming & Unix Fall 2018, CRN 44882, Oakland University Homework Assignment 6 - Using Arrays and Functions in C

DescriptionIn this final assignment, the students will demonstrate their ability to apply two ma

##### . The standard path finding involves finding the (shortest) path from an origin to a destination, typically on a map. This is an

Path finding involves finding a path from A to B. Typically we want the path to have certain properties,such as being the shortest or to avoid going t

##### . Develop a program to emulate a purchase transaction at a retail store. This program will have two classes, a LineItem class and a Transaction class. The LineItem class will represent an individual

Develop a program to emulate a purchase transaction at a retail store. Thisprogram will have two classes, a LineItem class and a Transaction class. Th

##### . SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of Sea Ports. Here are the classes and their instance variables we wish to define:

1 Project 1 Introduction - the SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of

##### . Project 2 Introduction - the SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of Sea Ports. Here are the classes and their instance variables we wish to define:

1 Project 2 Introduction - the SeaPort Project series For this set of projects for the course, we wish to simulate some of the aspects of a number of

Hire Me

Hire Me