Approximation Algorithms for Scheduling with Class Constraints

Jansen, Klaus; Lassota, Alexandra; Maack, Marten

doi:10.1145/3350755.3400247

Cited by 16 publications

(9 citation statements)

References 16 publications

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“…Faster algorithms for these block structured IPs thus immediately improve the running time for other problems. Applications include string algorithms [22], social choice games [23], scheduling [16,21,17], and bin packing problems [25].…”

Section: Related Resultsmentioning

confidence: 99%

Collapsing the Tower -- On the Complexity of Multistage Stochastic IPs

Klein¹,

Reuter²

2021

Preprint

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In this paper we study the computational complexity of solving a class of block structured integer programs (IPs) -so called multistage stochastic IPs. A multistage stochastic IP is an IP of the form max{c T x | Ax = b, l ≤ x ≤ u, x integral} where the constraint matrix A consists of small block matrices ordered on the diagonal line and for each stage there are larger blocks with few columns connecting the blocks in a tree like fashion. Over the last years there was enormous progress in the area of block structured IPs. For many of the known block IP classes -such as n-fold, tree-fold, and two-stage stochastic IPs, nearly matching upper and lower bounds are known concerning their computational complexity. One of the major gaps that remained however was the parameter dependency in the running time for an algorithm solving multistage stochastic IPs. Previous algorithms require a tower of t exponentials, where t is the number of stages, while only a double exponential lower bound was known. In this paper we show that the tower of t exponentials is actually not necessary. We can show an improved running time for the algorithm solving multistage stochastic IPs with a running time of 2• poly(d, n), where d is the sum of columns in the connecting blocks and n is the number of blocks on the lowest stage. Hence, we obtain the first bound by an elementary function for the running time of an algorithm solving multistage stochastic IPs. In contrast to previous works, our algorithm has only a triple exponential dependency on the parameters and only doubly exponential for every constant t. By this we come very close the known double exponential bound (based on the exponential time hypothesis) that holds already for two-stage stochastic IPs, i.e. multistage stochastic IPs with only two stages.The improved running time of the algorithm is based on new bounds for the proximity of multistage stochastic IPs. The idea behind the bound is based on generalization for a structural lemma originally used for two-stage stochastic IPs. While the structural lemma requires iteration to be applied to multistage stochastic IPs, our generalization directly applies to inherent combinatorial properties of multiple stages. Already a special case of our lemma yields an improved bound for the Graver Complexity of multistage stochastic IPs.

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Section: Related Resultsmentioning

confidence: 99%

Collapsing the Tower -- On the Complexity of Multistage Stochastic IPs

Klein¹,

Reuter²

2021

Preprint

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“…Scheduling is a broad topic that has been intensively studied. There are prior scheduling problem variants that consider reconfiguration in several forms [16,23,27,28,31,33,48], but none of them fully captures the characteristics of MIG. We elaborate the most relevant ones below (see more in §9).…”

Section: Related Scheduling Problemsmentioning

confidence: 99%

Serving DNN Models with Multi-Instance GPUs: A Case of the Reconfigurable Machine Scheduling Problem

Tan,

Li,

Zhang

et al. 2021

Preprint

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Multi-Instance GPU (MIG) is a new feature introduced by NVIDIA A100 GPUs that partitions one physical GPU into multiple GPU instances. With MIG, A100 can be the most costefficient GPU ever for serving Deep Neural Networks (DNNs). However, discovering the most efficient GPU partitions is challenging. The underlying problem is NP-hard; moreover, it is a new abstract problem, which we define as the Reconfigurable Machine Scheduling Problem (RMS).This paper studies serving DNNs with MIG, a new case of RMS. We further propose a solution, MIG-. MIGis an algorithm pipeline that blends a variety of newly designed algorithms and customized classic algorithms, including a heuristic greedy algorithm, Genetic Algorithm (GA), and Monte Carlo Tree Search algorithm (MCTS). We implement MIG-on Kubernetes. Our experiments show that compared to using A100 as-is, MIG-can save up to 40% GPUs while providing the same throughput.

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“…These ILPs often arise in the context of algorithmic applications. Probably the most famous one among such ILPs is the configuration ILP introduced by Gilmore and Gomory [10] and used for many packing and scheduling problems (e. g. [3,11,[17][18][19]).…”

Section: Note That This Implies That Sens(mentioning

confidence: 99%

Tightness of Sensitivity and Proximity Bounds for Integer Linear Programs

Berndt

Jansen

Lassota

2021

SOFSEM 2021: Theory and Practice of Computer Science

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We consider Integer Linear Programs (ILPs), where each variable corresponds to an integral point within a polytope P ⊆ R d , i. e., ILPs of the form min{c}. The distance between an optimal fractional solution and an optimal integral solution (called the proximity) is an important measure. A classical result by Cook et al. (Math. Program., 1986) shows that it is at most Δ Θ(d) where Δ = P ∩ Z d ∞ is the largest coefficient in the constraint matrix. Another important measure studies the change in an optimal solution if the right-hand side b is replaced by another right-hand side b . The distance between an optimal solution x w.r.t. b and an optimal solution x w.r.t. b (called the sensitivity) is similarly bounded, i. e., b−b 1 •Δ Θ(d) , also shown by Cook et al. (Math. Program., 1986).Even after more than thirty years, these bounds are essentially the best known bounds for these measures. While some lower bounds are known for these measures, they either only work for very small values of Δ, require negative entries in the constraint matrix, or have fractional right-hand sides. Hence, these lower bounds often do not correspond to instances from algorithmic problems. This work presents for each Δ > 0 and each d > 0 ILPs of the above type with non-negative constraint matrices such that their proximity and sensitivity is at least Δ Θ(d) . Furthermore, these instances are closely related to instances of the Bin Packing problem as they form a subset of columns of the configuration ILP. We thereby show that the results of Cook et al. are indeed tight, even for instances arising naturally from problems in combinatorial optimization.

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Approximation Algorithms for Scheduling with Class Constraints

Cited by 16 publications

References 16 publications

Collapsing the Tower -- On the Complexity of Multistage Stochastic IPs

Collapsing the Tower -- On the Complexity of Multistage Stochastic IPs

Serving DNN Models with Multi-Instance GPUs: A Case of the Reconfigurable Machine Scheduling Problem

Tightness of Sensitivity and Proximity Bounds for Integer Linear Programs

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