We describe a new dependent-rounding algorithmic framework for bipartite graphs. Given a fractional assignment $y$ of values to edges of graph $G = (U \cup V, E)$, the algorithms return an integral solution $Y$ such that each right-node $v \in V$ has at most one neighboring edge $f$ with $Y_f = 1$, and where the variables $Y_e$ also satisfy broad nonpositive-correlation properties. In particular, for any edges $e_1, e_2$ sharing a left-node $u \in U$, the variables $Y_{e_1}, Y_{e_2}$ have strong negative-correlation properties, i.e. the expectation of $Y_{e_1} Y_{e_2}$ is significantly below $y_{e_1} y_{e_2}$. This algorithm is a refinement of a dependent-rounding algorithm of Im \& Shadloo (2020) based on simulation of Poisson processes. Our algorithm allows greater flexibility, in particular, it allows ``irregular'' fractional assignments, and it gives more refined bounds on the negative correlation. 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 $1.407$-approximation for this problem. This significantly improves over the prior $1.45$-approximation ratio of Im & Li (2023).

翻译：本文描述了一种新的二部图依赖型舍入算法框架。给定图$G = (U \cup V, E)$边上值的分数分配$y$，该算法返回一个整数解$Y$，使得每个右节点$v \in V$至多有一条邻边$f$满足$Y_f = 1$，且变量$Y_e$同时满足广泛的非正相关性质。特别地，对于共享左节点$u \in U$的任意边$e_1, e_2$，变量$Y_{e_1}, Y_{e_2}$具有强负相关性质，即$Y_{e_1} Y_{e_2}$的期望显著低于$y_{e_1} y_{e_2}$。该算法是对Im与Shadloo（2020）基于泊松过程模拟的依赖型舍入算法的改进。我们的算法具有更高灵活性，尤其允许"不规则"分数分配，并提供更精细的负相关界。具有负相关性质的依赖型舍入方案已被用于非相关机器上最小化加权完成时间的作业调度近似算法（Bansal、Srinivasan与Svensson（2021），Im与Shadloo（2020），Im与Li（2023））。利用我们新的依赖型舍入算法，并辅以其他改进，我们对该问题实现了1.407-近似，显著优于Im与Li（2023）此前1.45-近似比。