Minimum-Norm Load Balancing Is (Almost) as Easy as Minimizing Makespan
We consider the minimum-norm load-balancing (MinNormLB) problem, wherein there are n jobs, each of which needs to be assigned to one of m machines, and we are given the processing times {p_{ij}} of the jobs on the machines. We also have a monotone, symmetric norm f:ℝ^m → ℝ_{≥ 0}. We seek an assignment σ of jobs to machines that minimizes the f-norm of the induced load vector load->_σ ∈ ℝ_{≥ 0}^m, where load_σ(i) = ∑_{j:σ(j) = i}p_{ij}. This problem was introduced by [Deeparnab Chakrabarty and Chaitanya Swamy, 2019], and the current-best result for MinNormLB is a (4+ε)-approximation [Deeparnab Chakrabarty and Chaitanya Swamy, 2019]. In the stochastic version of MinNormLB, the job processing times are given by nonnegative random variables X_{ij}, and jobs are independent; the goal is to find an assignment σ that minimizes the expected f-norm of the induced random load vector.
We obtain results that (essentially) match the best-known guarantees for deterministic makespan minimization (MinNormLB with 𝓁_∞ norm). For MinNormLB, we obtain a (2+ε)-approximation for unrelated machines, and a PTAS for identical machines. For stochastic MinNormLB, we consider the setting where the X_{ij}s are Poisson random variables, denoted PoisNormLB. Our main result here is a novel and powerful reduction showing that, for any machine environment (e.g., unrelated/identical machines), any α-approximation algorithm for MinNormLB in that machine environment yields a randomized α(1+ε)-approximation for PoisNormLB in that machine environment. Combining this with our results for MinNormLB, we immediately obtain a (2+ε)-approximation for PoisNormLB on unrelated machines, and a PTAS for PoisNormLB on identical machines. The latter result substantially generalizes a PTAS for makespan minimization with Poisson jobs obtained recently by [Anindya De et al., 2020].
Approximation algorithms
Load balancing
Minimum-norm optimization
LP rounding
Theory of computation~Approximation algorithms analysis
81:1-81:20
Track A: Algorithms, Complexity and Games
Supported in part by NSERC grant 327620-09 and an NSERC DAS Award.
Sharat
Ibrahimpur
Sharat Ibrahimpur
Department of Combinatorics and Optimization, University of Waterloo, Canada
https://orcid.org/0000-0002-1575-9648
Chaitanya
Swamy
Chaitanya Swamy
Department of Combinatorics and Optimization, University of Waterloo, Canada
https://orcid.org/0000-0003-1108-7941
10.4230/LIPIcs.ICALP.2021.81
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Sharat Ibrahimpur and Chaitanya Swamy
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