 
Summary: Online and Approximation Algorithms Fall Semester, 2011/12
Exercise 2: Dec 14, 2011
Lecturer: Prof. Yossi Azar
Write short but full and accurate answers. Each solution should appear on a separate page and each of its parts
should not exceed a page.
1. Consider the online load balancing problem on (even) m related machines. The speed of each of the first
m/2 machines is 1 and the speed of each of the other m/2 machines is 100. The goal is to minimize the
maximum load.
(a) Show a deterministic 2.01 competitive algorithm.
(b) Show a lower bound of 5/3 for any deterministic algorithm for any (even) m 6.
2. Consider the online load balancing problem of tasks on m machines in the restricted assignment model.
Consider the case where for each i the set of machines which is associated with job i is [1 . . . ki] for some
1 ki m. The goal is to minimize the maximum load.
(a) Design a constant competitive algorithm.
(b) Show a lower bound of 2 for deterministic algorithms, for any m 6, using only unit jobs. Hint: start
with m = 6. (you will a partial credit for a lower bound of 11/6.)
(c) Show a lower bound of 2O(1/m) for randomized algorithms, for any m, using only unit jobs.
3. Consider the online load balancing problem in the restricted assignment model where both the algorithm
and the optimum are allowed to split jobs (in any way).
(a) Define the natural "water level" algorithm and show that it is at most log m + 2 competitive.
