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：我们描述了一种新的针对二分图的依赖于取整算法框架。给定图 $G = (U \cup V, E)$ 中边赋值的分数向量 $\vec x$，该算法返回一个整数解 $\vec X$，使得每个右节点 $v \in V$ 至多有一条邻接边 $f$ 满足 $X_f = 1$，并且变量 $X_e$ 同时满足广泛的非正相关性。特别地，对于任意共享左节点 $u \in U$ 的两条边 $e_1, e_2$，变量 $X_{e_1}, X_{e_2}$ 具有强负相关性，即 $X_{e_1} X_{e_2}$ 的期望显著小于 $x_{e_1} x_{e_2}$。该算法基于生成负相关的指数随机变量，并在受 Im & Shadloo（2020）算法启发的冲突解决方案中使用它们。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于取整方案已被用于无关机调度中最小化加权完成时间的近似算法（Bansal, Srinivasan, & Svensson (2021), Im & Shadloo (2020), Im & Li (2023)）。利用我们新的依赖于取整算法，除其他改进外，我们获得了该问题的 $1.398$-近似比，显著优于 Im & Li（2023）此前给出的 $1.45$-近似比。