Load balancing of temporary tasks in the ℓ_p norm

We consider the on-line load balancing problem where there are m identical machines (servers). Jobs arrive at arbitrary times, where each job has a weight and a duration. A job has to be assigned upon its arrival to exactly one of the machines. The duration of each job becomes known only upon its termination (this is called temporary tasks of unknown durations). Once a job has been assigned to a machine it cannot be reassigned to another machine. The goal is to minimize the maximum over time of the sum (over all machines) of the squares of the loads, instead of the traditional maximum load. Minimizing the sum of the squares is equivalent to minimizing the load vector with respect to the ℓ₂ norm. We show that for the ℓ₂ norm the greedy algorithm performs within at most 1.493 of the optimum. We show (an asymptotic) lower bound of 1.33 on the competitive ratio of the greedy algorithm. We also show a lower bound of 1.20 on the competitive ratio of any algorithm. We extend our techniques and analyze the competitive ratio of the greedy algorithm with respect to the ℓ_p norm. We show that the greedy algorithm performs within at most 2 - Ω (1 / p) of the optimum. We also show a lower bound of 2 - O (ln p / p) on the competitive ratio of any on-line algorithm.