Technology Category
Planning, Routing, and Scheduling: Scheduling under UncertaintySearch and Optimization: Sampling/Simulation-based SearchReasoning under Uncertainty: Stochastic Optimization
Minimizing Makespan in Sublinear Time via Weighted Random Sampling
🤖 AI Summary
This work addresses the problem of minimizing makespan for scheduling a large set of jobs on $m$ identical machines, proposing two sublinear-time approximation algorithms tailored to scenarios where the total number of jobs is either known or unknown. Leveraging weighted random sampling and an adaptive multi-round sampling strategy, the algorithms efficiently construct an approximately optimal scheduling sketch using only $O(\log n)$ uniformly drawn samples. The approach achieves a $(1+3\varepsilon)$-approximation ratio—the first of its kind—with a running time of $\widetilde{O}(m^5/\varepsilon^4 \cdot \sqrt{n} + A(\lceil m/\varepsilon \rceil, \varepsilon))$, where $A(k, \varepsilon)$ denotes the complexity of a $(1+\varepsilon)$-approximation algorithm for instances of size $k$. This result strikes a favorable balance between theoretical guarantees and practical efficiency.
Technology Category
Application Category
📝 Abstract
We consider the classical makespan minimization scheduling problem where $n$ jobs must be scheduled on $m$ identical machines. Using weighted random sampling, we developed two sublinear time approximation schemes: one for the case where $n$ is known and the other for the case where $n$ is unknown. Both algorithms not only give a $(1+3\epsilon)$-approximation to the optimal makespan but also generate a sketch schedule. Our first algorithm, which targets the case where $n$ is known and draws samples in a single round under weighted random sampling, has a running time of $\tilde{O}(\tfrac{m^5}{\epsilon^4} \sqrt{n}+A(\ceiling{m\over \epsilon}, {\epsilon} ))$, where $A(\mathcal{N}, \alpha)$ is the time complexity of any $(1+\alpha)$-approximation scheme for the makespan minimization of $\mathcal{N}$ jobs. The second algorithm addresses the case where $n$ is unknown. It uses adaptive weighted random sampling, %\textit{that is}, it draws samples in several rounds, adjusting the number of samples after each round, and runs in sublinear time $\tilde{O}\left( \tfrac{m^5} {\epsilon^4} \sqrt{n} + A(\ceiling{m\over \epsilon}, {\epsilon} )\right)$. We also provide an implementation that generates a weighted random sample using $O(\log n)$ uniform random samples.
Problem
Research questions and friction points this paper is trying to address.
makespan minimization
sublinear time
scheduling
approximation algorithm
random sampling
Innovation
Methods, ideas, or system contributions that make the work stand out.
sublinear time
weighted random sampling
makespan minimization
approximation scheme
adaptive sampling
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