Stochastic Load Balancing on Unrelated Machines

Abstract

We consider the problem of makespan minimization: i.e., scheduling jobs on machines to minimize the maximum load. For the deterministic case, good approximations are known even when the machines are unrelated. However, the problem is not well-understood when there is uncertainty in the job sizes. In our setting the job sizes are stochastic, i.e., the size of a job j on machine i is a random variable X ij , whose distribution is known. (Sizes of different jobs are independent of each other.) The goal is to find a fixed assignment of jobs to machines, to minimize the expected makespan-i.e., the expected value of the maximum load over the m machines. For the identical machines special case when the size of a job is the same across all machines, a constant-factor approximation algorithm has long been known. However, the problem has remained open even for the next-harder related machines case. Our main result is a constant-factor approximation for the most general case of unrelated machines. The main technical challenge we overcome is obtaining an efficiently computable lower bound for the optimal solution. We give an exponential-sized LP that we argue gives a strong lower bound. Then we show how to round any fractional solution to satisfy only a small subset of the constraints, which are enough to bound the expected makespan of our solution. We then consider two generalizations. The first is the budgeted makespan minimization problem, where the goal is to minimize the makespan subject to scheduling any subset of jobs whose reward is at least some target reward R. We extend our above result to a constant-factor approximation here using polyhedral properties of the bipartite matching polytope. The second problem is the q-norm minimization problem, where we want to minimize the expected q -norm of the load vectors. Here we give an O(q/ log q)-approximation algorithm using a reduction to the deterministic q-norm problem with side constraints.

Stochastic ℓp Load Balancing and Moment Problems via the L-Function Method

“…This classic problem has been widely studied in its stochastic [KRT00, GI99, GKNS18, Pin04], deterministic [LST90, AERW04, AE05, KMPS09,MS14], and online versions [AAG + 95, AAS01, BCK00, Car08, CFK + 11, Mol17]. See [GKNS18] for a comprehensive discussion and literature review on stochastic load balancing, most relevant for us. The deterministic versions of such problems can typically be well-approximated through the use of convex programs; for example, this method has provided constant-factor approximations for the deterministic version of STOCHLOADBAL p [AE05, KMPS09,MS14].…”

“…They also use it to provide approximations for stochastic bin-packing and knapsack problems (all packing-or ℓ ∞ -type problems). Only recently, Gupta et al [GKNS18] managed to use this fruitful notion to obtain a constant approximation for the unrelated machines case (but still p = ∞). However, suitable notions of effective size have not been used for p-power-type functions.…”

“…However, suitable notions of effective size have not been used for p-power-type functions. For example, for STOCHLOADBAL p with general p only an O( p ln p )-approximation is known, also due to [GKNS18], and relies on other techniques (expected size as a proxy plus Rosenthal's Inequality). Oddly, this approximation ratio goes to infinity as p → ∞, despite the constant-factor approximation known for p = ∞.…”

Stochastic Makespan Minimization in Structured Set Systems (Extended Abstract)

“…The "volume" constraints in our LP also has some similarity to those in [12]: however, a key difference here is that the random variables loading different resources are correlated (whereas they were independent in [12]). Indeed, this is why our LP can only be solved approximately whereas the LP relaxation in [12] was optimally solvable. We emphasize that our main contribution is the rounding algorithm ideas uses a new set of ideas; these lead to the O(log log m) approximation bound, whereas the rounding in [12] obtained a constant-factor approximation.…”

“…Indeed, this is why our LP can only be solved approximately whereas the LP relaxation in [12] was optimally solvable. We emphasize that our main contribution is the rounding algorithm ideas uses a new set of ideas; these lead to the O(log log m) approximation bound, whereas the rounding in [12] obtained a constant-factor approximation. Note that we also prove a super-constant integrality gap in our setting, even for the case of intervals in a line.…”