Many optimization and scheduling problems can be abstracted in terms of a bipartite ``assignment graph" $G = (L \cup R, E)$, where the goal is to select exactly one edge for each right-node. For example, a right-node may correspond to a job, and a left-node to a possible machine assignment. A common strategy to solve such problems is to obtain a fractional relaxation $x_e$ for each edge $e$, and then have each right-node independently select an edge with probability $x_e$. However, this may cause the left-nodes to become unevenly loaded, leading to suboptimal solutions for some problems. To address this, a number of algorithms for dependent rounding with strong negative correlation have been developed, e.g. Bansal, Srinivasan & Svensson (2021), Im & Shadloo (2020), Im & Li (2023), Harris (2024), Naor, Srinivasan & Wajc (2025). We introduce a new method for this, which we call the \emph{Dirichlet mechanism}. It is based on having each left-node draw Dirichlet random variables for its edges, and then having each right-node select an edge based on these values. This achieves quantitatively stronger negative correlation than previous algorithms, and is also simpler since it avoids the need for a tie-breaking mechanism. We illustrate the mechanism with improved approximation ratios for two problems. For oblivious online dependent rounding, we achieve a $0.68$-approximation which improves upon the previous $0.652$-approximation of Naor, Srinivasan & Wajc (2025). For the problem of scheduling jobs on unrelated machines to minimize weighted completion time, we achieve a $1.387$-approximation which improves upon the $1.398$-approximation of Harris (2024). (A recent algorithm of Li (2025) based on iterated rounding also provides a $1.36$-approximation if the weights of each job are independent of machine.)

翻译：许多优化与调度问题可抽象为二分“指派图” $G = (L \cup R, E)$，其目标是为每个右节点恰好选择一条边。例如，右节点可对应作业，左节点对应可能的机器分配。解决此类问题的常见策略是，先为每条边 $e$ 获取分数松弛 $x_e$，然后每个右节点独立地以概率 $x_e$ 选择一条边。然而，这可能导致左节点负载不均，从而在某些问题中产生次优解。为此，研究者已开发了一系列具有强负相关性的相依舍入算法，例如 Bansal、Srinivasan 与 Svensson (2021)、Im 与 Shadloo (2020)、Im 与 Li (2023)、Harris (2024)、Naor、Srinivasan 与 Wajc (2025)。我们提出一种新方法——狄利克雷机制。该方法基于：每个左节点为其连接的边生成狄利克雷随机变量，每个右节点再根据这些值选择一条边。与现有算法相比，该机制在量化上实现了更强的负相关性，且因无需平局处理机制而更加简洁。我们通过两个问题的改进近似比展示了该机制。对于遗忘在线相依舍入，我们实现了 $0.68$ 的近似比，优于 Naor、Srinivasan 与 Wajc (2025) 的 $0.652$ 近似比。对于不相关机器上最小化加权完工时间的作业调度问题，我们实现了 $1.387$ 的近似比，优于 Harris (2024) 的 $1.398$ 近似比。（若每个作业的权重与机器独立，Li (2025) 近期基于迭代舍入的算法也可达到 $1.36$ 的近似比。）