A Simple Approximation Algorithm for Vector Scheduling and Applications to Stochastic Min-Norm Load Balancing
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We consider the Vector Scheduling problem on identical machines: we have m machines, and a set J of n jobs, where each job j has a processing-time vector p_j∈ℝ^d_≥ 0. The goal is to find an assignment σ:J→ [m] of jobs to machines so as to minimize the makespan max_i∈ [m]max_r∈ [d]( ∑_j:σ(j)=ip_j,r). A natural lower bound on the optimal makespan is lb :=max{max_j∈ J,r∈ [d]p_j,r,max_r∈ [d](∑_j∈ Jp_j,r/m)}. Our main result is a very simple O(log d)-approximation algorithm for vector scheduling with respect to the lower bound lb: we devise an algorithm that returns an assignment whose makespan is at most O(log d)*lb. As an application, we show that the above guarantee leads to an O(log log m)-approximation for Stochastic Minimum-Norm Load Balancing (StochNormLB). In StochNormLB, we have m identical machines, a set J of n independent stochastic jobs whose processing times are nonnegative random variables, and a monotone, symmetric norm f:ℝ^m →ℝ_≥ 0. The goal is to find an assignment σ:J→ [m] that minimizes the expected f-norm of the induced machine-load vector, where the load on machine i is the (random) total processing time assigned to it. Our O(log log m)-approximation guarantee is in fact much stronger: we obtain an assignment that is simultaneously an O(log log m)-approximation for StochNormLB with all monotone, symmetric norms. Next, this approximation factor significantly improves upon the O(log m/log log m)-approximation in (Ibrahimpur and Swamy, FOCS 2020) for StochNormLB, and is a consequence of a more-general black-box reduction that we present, showing that a γ(d)-approximation for d-dimensional vector scheduling with respect to the lower bound lb yields a simultaneous γ(log m)-approximation for StochNormLB with all monotone, symmetric norms.
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