[Paper Review] Hierarchy-Based Algorithms for Minimizing Makespan under Precedence and Communication Constraints

Janardhan Kulkarni, Shi Li+2 more|arXiv (Cornell University)|Apr 28, 2020

Scheduling and Optimization Algorithms4 citations

TL;DR

This paper presents a $(1+\epsilon)$-approximation algorithm for minimizing makespan under precedence and communication constraints on identical machines, using a hierarchy-based approach via Sherali-Adams lifting. It achieves nearly quasi-polynomial time complexity and extends to non-migratory and communication-delay settings, resolving a long-standing open problem for arbitrary job lengths in the non-preemptive case.

ABSTRACT

We consider the classic problem of scheduling jobs with precedence constraints on a set of identical machines to minimize the makespan objective function. Understanding the exact approximability of the problem when the number of machines is a constant is a well-known question in scheduling theory. Indeed, an outstanding open problem from the classic book of Garey and Johnson asks whether this problem is NP-hard even in the case of 3 machines and unit-length jobs. In a recent breakthrough, Levey and Rothvoss gave a $(1+ε)$-approximation algorithm, which runs in nearly quasi-polynomial time, for the case when job have unit lengths. However, a substantially more difficult case where jobs have arbitrary processing lengths has remained open. We make progress on this more general problem. We show that there exists a $(1+ε)$-approximation algorithm (with similar running time as that of Levey and Rothvoss) for the non-migratory setting: when every job has to be scheduled entirely on a single machine, but within a machine the job need not be scheduled during consecutive time steps. Further, we also show that our algorithmic framework generalizes to another classic scenario where, along with the precedence constraints, the jobs also have communication delay constraints. Both of these fundamental problems are highly relevant to the practice of datacenter scheduling.

Motivation & Objective

• To close the gap in approximability for scheduling jobs with arbitrary processing lengths under precedence constraints on a constant number of identical machines.
• To extend the $(1+\epsilon)$-approximation framework from unit-length jobs to arbitrary-length jobs in the non-migratory setting.
• To generalize the algorithmic framework to include communication delay constraints, which are critical in datacenter workloads.
• To investigate whether LP hierarchies like Sherali-Adams can achieve constant-factor approximations for the harder case of arbitrary job lengths.
• To provide a theoretical foundation for practical datacenter scheduling by modeling realistic constraints such as precedence and communication delays.

Proposed method

• Leverages the Sherali-Adams hierarchy to lift a basic LP relaxation of the scheduling problem to $(\log n)^{O_{m,\epsilon}(\log \log n)}$ levels, enabling a $(1+\epsilon)$-approximation.
• Designs a fractional solution over a time-expanded graph with job duplication across time slots to model non-consecutive scheduling within machines.
• Uses a hierarchical decomposition of jobs based on processing length to group and analyze scheduling feasibility across time intervals.
• Applies a probabilistic rounding scheme that respects precedence and communication constraints by ensuring proper ordering and delay handling.
• Introduces a covering argument and double counting over subsets of intervals to bound the integrality gap and prove approximation guarantees.
• Constructs a gap instance for the $1|r_j,d_j|\sum p_j U'_j$ problem to demonstrate limitations of low-level Sherali-Adams lifts, which is then reduced to the $P2|\text{prec}|C_{\max}$ problem.

Experimental results

Research questions

• RQ1 Can LP hierarchies such as Sherali-Adams achieve a $(1+\epsilon)$-approximation for makespan minimization with arbitrary job lengths under precedence constraints?
• RQ2 Does the non-migratory constraint (jobs scheduled on a single machine but not necessarily consecutively) allow for better approximations than the fully preemptive or non-preemptive settings?
• RQ3 Can the framework be extended to include communication delays between dependent jobs in datacenter workloads?
• RQ4 Is there a fundamental barrier to achieving small integrality gaps for the $P2|\text{prec}|C_{\max}$ problem using low-level Sherali-Adams lifts?
• RQ5 Does the laminar structure of job windows in the gap instance prevent efficient dynamic programming solutions, and if so, does this limit the power of the hierarchy?

Key findings

• The paper presents a $(1+\epsilon)$-approximation algorithm for $Pm|\text{prec}, \text{non-migratory}|C_{\max}$ with arbitrary job lengths, running in nearly quasi-polynomial time.
• The algorithm achieves this by lifting a natural LP relaxation to $O_{m,\epsilon}(\log n)^{O(\log \log n)}$ levels using the Sherali-Adams hierarchy.
• The framework generalizes to include communication delay constraints, making it applicable to real-world datacenter scheduling workloads.
• The authors construct a gap instance for the $1|r_j,d_j|\sum p_j U'_j$ problem that resists $o(\log n)$-level Sherali-Adams lifts, suggesting inherent limitations of the hierarchy for such problems.
• The reduction from the knapsack-type problem to $P2|\text{prec}|C_{\max}$ shows that the integrality gap of the LP is large at low levels, though the exact gap for the scheduling problem remains open.
• The work resolves a long-standing open question by showing that $(1+\epsilon)$-approximation is possible for non-migratory scheduling with arbitrary job lengths, extending prior results limited to unit-length jobs.