Algorithmic Methods Fall Semester, 2010/11 Exercise 3: December 20, 2010 Summary: Algorithmic Methods Fall Semester, 2010/11 Exercise 3: December 20, 2010 Lecturer: Prof. Yossi Azar Write short but full and accurate answers. Each question should start on a new separate page and each of its parts should not exceed a page. 1. We are given n jobs and m unrelated machines. The load of job i on machine j is wij. The load of a machine is the sum of the weights of the jobs assigned to it. Each job i should be either assigned to some machine or we encounter a penalty of pi. The goal is to minimize the maximum load plus the total penalty of the unassigned jobs. Design a 2.62 approximation algorithm for the problem. Hint: Design first a 3 approximation algorithm. 2. Suppose we are given a regular graph G = (V, E) of degree . Each vertex has a different i.d (which initially is unknown to the others) between 0 to 2n - 1 where |V | = n. Recall that a local algorithm with k rounds is an algorithm where each vertex decides on its output after k synchronized communication rounds with its neighbors. Find a local algorithm that colors the graph in + 1 colors in log n + 2O() rounds. Remark: a solution in log n + 2O( log ) rounds will receive almost all points. 3. Assume we are given a rooted tree where vertices may have arbitrary degrees. Each vertex has a unique label between 1 and n. Each vertex has to choose a color based only on local Collections: Computer Technologies and Information Sciences