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