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：我们提出了一种新的针对二分图的依赖舍入算法框架。给定图 $G = (U \cup V, E)$ 边上值的一个分数分配 $\vec x$，该算法返回一个整数解 $\vec X$，使得每个右侧节点 $v \in V$ 至多有一条邻边 $f$ 满足 $X_f = 1$，并且变量 $X_e$ 还满足广泛的非正相关性质。特别地，对于共享左侧节点 $u \in U$ 的任意两条边 $e_1, e_2$，变量 $X_{e_1}, X_{e_2}$ 具有强负相关性质，即 $X_{e_1} X_{e_2}$ 的期望值显著低于 $x_{e_1} x_{e_2}$。该算法基于生成负相关的指数随机变量，并将其用于一种受 Im & Shadloo (2020) 算法启发的争用解决方案中。我们的算法提供了更强且灵活得多的负相关性质。具有负相关性质的依赖舍入方案已被用于无关机器上作业调度以最小化加权完工时间的近似算法中（Bansal, Srinivasan, & Svensson (2021), Im & Shadloo (2020), Im & Li (2023)）。利用我们新的依赖舍入算法，以及其他改进，我们为此问题获得了 $1.398$ 的近似比。这显著改进了 Im & Li (2023) 先前 $1.45$ 的近似比。