We describe a new dependent-rounding algorithmic framework for bipartite graphs. Given a fractional assignment $\vec x$ of values to edges of graph $G = (U \cup V, E)$, the algorithms return an integral solution $\vec X$ such that each right-node $v \in V$ has at most one neighboring edge $f$ with $X_f = 1$, and where the variables $X_e$ also satisfy broad nonpositive-correlation properties. In particular, for any edges $e_1, e_2$ sharing a left-node $u \in U$, the variables $X_{e_1}, X_{e_2}$ have strong negative-correlation properties, i.e. the expectation of $X_{e_1} X_{e_2}$ is significantly below $x_{e_1} x_{e_2}$. This algorithm is based on generating negatively-correlated Exponential random variables and using them in a contention-resolution scheme inspired by an algorithm Im & Shadloo (2020). Our algorithm gives stronger and much more flexible negative correlation properties. Dependent rounding schemes with negative correlation properties have been used for approximation algorithms for job-scheduling on unrelated machines to minimize weighted completion times (Bansal, Srinivasan, & Svensson (2021), Im & Shadloo (2020), Im & Li (2023)). Using our new dependent-rounding algorithm, among other improvements, we obtain a $1.398$-approximation for this problem. This significantly improves over the prior $1.45$-approximation ratio of Im & Li (2023).

翻译：我们描述了一种针对二分图的新型依赖舍入算法框架。给定图 $G = (U \cup V, E)$ 边上数值的分数分配 $\vec x$，该算法返回一个整数解 $\vec X$，使得每个右节点 $v \in V$ 至多有一条相邻边 $f$ 满足 $X_f = 1$，且变量 $X_e$ 同时满足广泛的非正相关性质。特别地，对于任意共享左节点 $u \in U$ 的边 $e_1, e_2$，变量 $X_{e_1}, X_{e_2}$ 具有强负相关性质，即 $X_{e_1} X_{e_2}$ 的期望显著低于 $x_{e_1} x_{e_2}$。该算法基于生成负相关的指数随机变量，并将其用于受 Im & Shadloo (2020) 算法启发的竞争解决框架。我们的算法提供了更强且更灵活的负相关性质。具有负相关性质的依赖舍入方案已被用于非关联机器上最小化加权完工时间的作业调度近似算法（Bansal, Srinivasan, & Svensson (2021)，Im & Shadloo (2020)，Im & Li (2023)）。通过使用我们新的依赖舍入算法，并辅以其他改进，我们获得了该问题的 $1.398$ 近似比。这显著优于 Im & Li (2023) 先前得到的 $1.45$ 近似比。