Multi-resource Fair Allocation with Bounded Number of Tasks in Cloud Computing Systems

We study a problem of multi-resource fair allocation with bounded number of tasks, and propose the lexicographically max-min dominant share (LMMDS) fair allocation mechanism, which is a generalization of the popular dominant resource fairness (DRF) mechanism. We prove that LMMDS satisfies sharing incentives, group strategyproofness, envy-freeness, and Pareto efficiency, by exploiting the properties of the optimal solution. In addition, we design a non-trivial optimal algorithm to find a LMMDS fair allocation whose running time is linear in the number of users n, when the number of types of resources m is bounded. Finnally, we prove that the approximation ratio of LMMDS (or DRF) is infinity for the general case, and exactly n for a special case, improving the previous best lower bound m.