COMP 360: Algorithm Design Instructor: Yang Cai Problem Set 4 Due: Apr 1, 2015 General Rules: Follow the collaboration policy. In particular, each of you should write your own solution even if you have collaborated with your classmates. In solving these questions you may consult books but you may not search on the web for solutions. You should drop your solutions in the assignment drop-off box located in the Trottier Building. No late assignments accepted. 1. (10 pts) Backtracking: Vertex Coloring. We can use the following state-space tree T for a backtracking algorithm to test whether a graph is 3-colorable. The root node of T gives vertex A color 1. Every node in the tree then has three children corresponding to each of the three colors. For example, the root has three children that correspond to giving vertex B the colors 1, 2 and 3 respectively. These three nodes each have three children corresponding to the color given to node C etc. Apply the backtracking algorithm to test whether or not the following graph is 3-colorable. [You should also start with the root node A, order the children alphabetically, and search using DFS order.] B A C G H D F E 2. (20 pts) Local Search: Load Balancing p704, Qu. 12.4. 3. (10 pts) Given a set P of n points on the plane, consider the problem of finding the smallest circle containing all the points in P . Show that the following is a 2-factor approximation algorithm for this problem. Pick a point x ∈ P , and set r to be the distance of the farthest point in P from x. Output the circle centered at x with radius r. 4. (15 pts)Knapsack: Let W be the capacity, wi ≤ W be item i’s weight and vi be item i’s value. Consider the following algorithm. 2-1 (a) Reorder the items according to their values and weights, so that v2 vn v1 ≥ ≥ ··· ≥ . w1 w2 wn (b) Pick items in this order until one doesn’t fit, and then halt. (c) Compare the value of the solution from step (b) and the largest value of a singe item, return which one is higher (either the solution of step (b) or the largest vi ). Show that this is a 2-approximation algorithm for the knapsack problem. 5. (20 pts) TSP: We define two versions of the Traveling Salesman Problem: • TSP with repetitions (TSP-R): The costs do not necessarily satisfy triangle inequality. Find a min cost cycle that visits every vertex at least once and returns to the origin, where the cycle can visit some vertex more than once. • ∆-TSP: The costs satisfy triangle inequality. Find a min cost cycle that visits every vertex exactly once and returns to the origin. Argue that these two versions are equivalent. That is, for every c ≥ 1, there is a polynomial time c-approximate algorithm for ∆-TSP if and only if there is a polynomial time c-approximate approximation algorithm for TSP-R. 2-2