# Homework Assignments

Homework is due the Tuesday after the work is assigned. Remember that you are allowed to turn in up to 4 homework assignments late, but they should not be more than 1 week late in any case.

#### The Assignments:

• Homework 1, due 02/06/01.

• Homework 2, due 02/13/01.

• Homework 3, due 02/20/01.

• Homework 4, due 02/27/01.
• Exercise 21.4-1, page 438.
• Problem 21-2, page 439.
• Problem 22-1, page 458.

• Homework 5, due 03/06/01.
• Exercise 22.3-4, page 450.
Note: where the question mentions the UNION operation, replace it with the LINK operation.
• Problem 22-3, page 460.

• Homework 6, due 03/13/01.
• Exercise 27.2-8, page 600.
• Exercise 27.2-9, page 600.
• Problem 27-1, page 625.

• Homework 7, due 04/03/01.
• Exercise 30.1-1, page 700.
• Exercise 30.1-3, page 700.
• Problem 30-1, page 726.

• Homework 8, due 04/10/01.
• Exercise 28.1-2, page 638.
• Exercise 28.4-3, page 648.
• Exercise 28.5-4, page 650.

• Homework 9, due 04/17/01.
• Exercise 6.3-3, page 114.
• Exercise 6.4-2, page 119.
• Consider the following modification to the randomized MinCut algorithm presented in class. Instead of choosing a random edge for contraction, choose two vertices at random and merge them into a single vertex. Here the two vertices do not have to be connected by an edge. Show that there are inputs on which the probability that this modified algorithm finds a min-cut is exponentially small. [Exercise 1.2, Randomized Algorithms, Motwani and Raghavan.]

• Homework 10, due 04/24/01.
• Exercise 35.1-3, page 891.
• Exercise 35.1-5, page 891.
• Exercise 35.2-6, page 898.

• Homework 11, due 05/01/01.
• Exercise 35.3-6, page 908.
• Exercise 35.4-1, page 912.
• Exercise 35.4-4, page 912.

• Homework 12, due 05/08/01.
• Exercise 36.5-2, page 960.
• Exercise 36.5-5, page 961.
• Prove that 3-colorability for planar graphs is NP-complete. (You may assume that 3-colorability for general graphs is NP-complete.)

• Homework 13, due 05/15/01.
• Exercise 36.2-3, page 928.
• Exercise 36.5-4, page 960.
• Prove that Steiner Tree on Graphs (STG) is NP-complete. (Hint: reduce from 3DM.)

STG = { (G,R,k) | where G=(V,E) is an undirected graph, R is a subset of V and k <= |V|-1 such that there exists a subtree T of G that includes all the vertices of R and T contains no more than k edges.}