Partition with Side Effects

In data centers, many tasks (services, virtual machines or computational jobs) share a single physical machine. We propose a new resource management model for such colocation. Our model uses two parameters of a task -- its size and its type -- to characterize how a task influences the performance of the other tasks allocated on the same machine. As typically a data~center hosts many similar, recurring tasks (e.g.: a webserver, a database, a CPU-intensive computation), the resource manager should be able to construct these types and their performance interactions. Moreover, realistic variants of our model are polynomially-solvable, in contrast to the NP-hard vector packing used previously. In particular, we minimize the total cost in a model in which each task's cost is a function of the total sizes of tasks allocated on the same machine (each type is counted separately). We show that for a linear cost function the problem is strongly NP-hard, but polynomially-solvable in some particular cases. We propose an algorithm polynomial in the number of tasks (but exponential in the number of types and machines), and another algorithm polynomial in the number of tasks and machines (but exponential in the number of types and admissible sizes of tasks). When there is a single type, we give a polynomial time algorithm. We also prove that, even for a single type, the problem becomes NP-hard for convex costs.