Assigning sporadic tasks to unrelated machines

We study the problem of assigning sporadic tasks to unrelated machines such that the tasks on each machine can be feasibly scheduled. Despite its importance for modern real-time systems, this problem has not been studied before. We present a polynomial-time algorithm which approximates the problem with a constant speedup factor of 8+26‾√≈12.9 and show that any polynomial-time algorithm needs a speedup factor of at least 2, unless P = NP. In the case of a constant number of machines we give a polynomial-time approximation scheme. Key to these results are two new relaxations of the demand bound function, the function that yields a sufficient and necessary condition for a task system on a single machine to be feasible. In particular, we present new methods to approximate this function to obtain useful structural properties while incurring only bounded loss in the approximation quality. For the constant speedup result we employ a very general rounding procedure for linear programs (LPs) which model assignment problems with capacity-type constraints. It ensures that the cost of the rounded integral solution is no more than the cost of the optimal fractional LP solution and the capacity constraints are violated only by a bounded factor, depending on the structure of the matrix that defines the LP. In fact, our rounding scheme generalizes the well-known 2-approximation algorithm for the generalized assignment problem due to Shmoys and Tardos.