贝叶斯调度中的间隙与增强
Gaps and Augmentations in Bayesian Scheduling
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中文总结 AI 辅助
本研究探讨贝叶斯调度中先验信息、随机化与资源增强对真机制近似比的影响，证明随机化BIC机制存在4/3下界及8/7完整性间隙，并给出MinWork机制在特定条件下O(m/ε)采样层实现1+ε近似的资源增强结果。
中文摘要 AI 辅助
最近对Nisan--Ronen猜想的解决确立了确定性真机制在m台不相关机器上最小化完工时间的最坏情况近似比的最优值恰好为m。我们探究先验信息、贝叶斯激励相容性（BIC）、随机化以及机器侧资源增强如何改变这一障碍。我们获得三个主要结果。首先，我们证明对于两台机器，随机化BIC机制存在一个渐进的4/3下界，强化了之前确定性BIC的1.2下界。其次，在无先验的两台机器设置中，随机化将已知的真机制近似比从2改进到7/4。我们证明随机化BIC调度机制同样严格强于其确定性对应物，展示了一个渐进的BIC完整性间隙为8/7。因此，在先验依赖的BIC调度中，最优性可以严格需要随机化。这与多维收益最大化中彩票的作用相呼应，其中随机化可以严格提高收益。第三，我们证明当任务分配在机器间足够均匀分布（更正式地说，当两两碰撞参数Δ_r被一个独立于机器数量m和采样层数r的常数所界定）时，标准真机制MinWork的O(m/ε)个采样层足以实现对原始第一最优基准的1+ε近似。等价地，每台机器O(m/ε)个采样副本就足够了。这一资源增强结果与Bulow--Klemperer视角相呼应。作为副产品，在标准的无增强模型中，MinWork对每个先验都能实现一个(1+Δ_1(m-1)/2)近似于第一最优基准。
英文摘要
The recent resolution of the Nisan--Ronen conjecture~\cite{NR,CKK} establishes that the optimal worst-case approximation ratio of deterministic truthful mechanisms for makespan on m unrelated machines is exactly m. We ask how prior information, Bayesian incentive compatibility (BIC), randomization, and machine-side resource augmentation change this barrier. We obtain three main results. First, we prove an asymptotic 4/3 lower bound for randomized BIC mechanisms with two machines, strengthening the previous 1.2 deterministic-BIC lower bound~\cite{MS}. Second, in the prior-free setting for two machines, randomization improves the known truthful ratio from 2 to 7/4~\cite{NR}. We show that randomized BIC scheduling mechanisms are likewise strictly more powerful than their deterministic counterparts, exhibiting an asymptotic BIC-integrality gap of 8/7. Thus, optimality in prior-dependent BIC scheduling can strictly require randomization. This parallels the role of lotteries in multidimensional revenue maximization, where randomization can strictly improve revenue~\cite{MV,BCKW}. Third, we show that when job assignments are sufficiently well spread across machines (more formally, when the pairwise collision parameter $Δ_r$ is bounded by a constant independent of both the number of machines m and the number of sampled layers r), then $O(m/\varepsilon)$ sampled layers suffice for the standard truthful MinWork mechanism to achieve a $1+\varepsilon$-approximation to the original first-best benchmark. Equivalently, $O(m/\varepsilon)$ sampled replicas per machine suffice. This resource-augmentation result parallels the Bulow--Klemperer perspective~\cite{BK,EFFTW}. As a by-product, in the standard unaugmented model, MinWork achieves a $(1+\frac{Δ_1(m-1)}{2})$-approximation to the first-best benchmark for every prior.
