改进的二范数在线负载均衡
Improved Online Load Balancing in the Two-Norm
AI总结：
本文针对无关机器上的在线负载均衡问题，提出首个突破竞争比5的算法，达到4.9843，并通过新的原始-对偶框架统一证明了多个相关算法的最优性。
摘要：
我们研究无关机器上的在线负载均衡问题，目标是最小化机器负载的 $\ell_2$ 范数的平方。Awerbuch 等人（STOC'95）的贪心算法对于确定性算法是最优的，其竞争比为 $3 + 2 \sqrt{2} \approx 5.828$，而 Caragiannis（SODA'08）给出了一个基于独立舍入的改进的 $5$-竞争随机算法。在本工作中，我们提出了第一个突破竞争比 $5$ 这一障碍的算法，达到了 $4.9843$ 的界。为得到这一结果，我们使用一个新的原始-对偶框架，基于一个自然的半定规划松弛来分析该问题，并结合 Im 和 Shadloo（SODA'20）的相关随机舍入过程的在线实现。这一新的原始-对偶框架还给出了 $(3 + 2 \sqrt{2})$-竞争贪心算法、$5$-竞争随机独立舍入算法以及一个新的 $4$-竞争最优分数算法的竞争比的新的、简单且统一的证明。我们还提供了下界，表明先前最好的随机算法在独立舍入算法中是最优的，我们的新分数算法是最优的，并且一个简单的贪心算法对于密切相关的在线调度问题 $R || \sum w_j C_j$ 是最优的。
英文摘要：
We study the online load balancing problem on unrelated machines, with the objective of minimizing the square of the $\ell_2$ norm of the loads on the machines. The greedy algorithm of Awerbuch et al. (STOC'95) is optimal for deterministic algorithms and achieves a competitive ratio of $3 + 2 \sqrt{2} \approx 5.828$, and an improved $5$-competitive randomized algorithm based on independent rounding has been shown by Caragiannis (SODA'08). In this work, we present the first algorithm breaking the barrier of $5$ on the competitive ratio, achieving a bound of $5-c$ for some $c \geq 0.0157$. To obtain this result, we use a new primal-dual framework to analyze this problem based on a natural semidefinite programming relaxation, together with an online implementation of a correlated randomized rounding procedure of Im and Shadloo (SODA'20). This novel primal-dual framework also yields new, simple and unified proofs of the competitive ratio of the $(3 + 2 \sqrt{2})$-competitive greedy algorithm, the $5$-competitive randomized independent rounding algorithm, and that of a new $4$-competitive optimal fractional algorithm. We also provide lower bounds showing that the previous best randomized algorithm is optimal among independent rounding algorithms, that our new fractional algorithm is optimal, and that a simple greedy algorithm is optimal for the closely related online scheduling problem $R || \sum w_j C_j$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
