We describe new dependent-rounding algorithms for bipartite graphs. Given a fractional matching $x$ of graph $G = (U \cup V, E)$, the algorithms return an integral solution $X$ such that each right-node $v \in V$ has at most one edge, and where the variables $X_e$ also satisfy broad non-positive 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 \emph{strong} negative-correlation, i.e. the expectation of $X_{e_1} X_{e_2}$ is significantly below $x_{e_1} x_{e_2}$. Dependent rounding schemes with these properties have been used for a approximation algorithms for job-scheduling on unrelated machines to minimize weighted completion times, among other applications. Our new algorithm achieves simpler and qualitatively stronger bounds compared to prior algorithms. In particular, we achieve a negative-correlation property $$ \E[X_{e_1} X_{e_2}] \leq 0.79751 \ x_{e_1} x_{e_2}, $$ which is a significant constant-factor improvement over Baveja, Qu & Srinivasan (2023).

翻译：我们描述了二分图的新依赖舍入算法。给定图 $G = (U \cup V, E)$ 的分数匹配 $x$，该算法返回一个整数解 $X$，使得每个右节点 $v \in V$ 最多有一条边，并且变量 $X_e$ 还满足广泛的非正相关性质。特别地，对于共享左节点 $u \in U$ 的任意边 $e_1, e_2$，变量 $X_{e_1}, X_{e_2}$ 具有强负相关，即 $X_{e_1} X_{e_2}$ 的期望显著低于 $x_{e_1} x_{e_2}$。具有这些性质的依赖舍入方案已被用于无关机上的作业调度近似算法（以最小化加权完成时间）等应用。与先前的算法相比，我们的新算法实现了更简单且定性更强的界。特别地，我们实现了负相关性质 $$ \E[X_{e_1} X_{e_2}] \leq 0.79751 \ x_{e_1} x_{e_2}, $$ 这是相对于 Baveja、Qu 和 Srinivasan (2023) 的显著常因子改进。