Bi-objective Optimization: An Online Algorithm for Job Assignment

We study an online problem that occurs when the capacities of machines are heterogeneous and all jobs are identical. Each job is associated with a subset, called feasible set, of the machines that can be used to process it. The problem involves assigning each job to a single machine in its feasible set, i.e., to find a feasible assignment. The objective is to maximize the throughput, which is the sum of the bandwidths of the jobs; and minimize the total load, which is the sum of the loads of the machines. In the online setting, the jobs arrive one-by-one and an algorithm must make decisions based on the current state without knowledge of future states. By contrast, in the offline setting, all the jobs with their feasible sets are known in advance to an algorithm. Let m denote the total number of machines, α denote the competitive ratio with respect to the throughput and β denote the competitive ratio with respect to the total load. In this paper, our contribution is that we propose an online algorithm that finds a feasible assignment with a throughput-competitive upper bound $\alpha=O(\sqrt{m})$, and a total-load-competitive upper bound $\beta=O(\sqrt{m})$. We also show a lower bound $\alpha\beta=\Omega(\sqrt{m})$ of the problem in the offline setting, which implies a lower bound $\alpha\beta=\Omega(\sqrt{m})$ of the problem in the online setting.