A note on rounding fractional matchings with constant-factor strong negative correlation

Abstract

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 *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 $$\mathbb{E}[X_{e_1}{X}_{e_2}]\le 0.79751x_{e_1}{x}_{e_2},$$ which is a significant constant-factor improvement over Baveja, Qu & Srinivasan (2023).