The Dirichlet Mechanism for Rounding with Strong Negative Correlation, with Applications

Abstract

Many optimization and scheduling problems can be abstracted in terms of a bipartite "assignment graph" G = (L ∪ R, E), where the goal is to select exactly one edge for each right-node. For example, a right-node may correspond to a job, and a left-node to a possible machine assignment. A common strategy to solve such problems is to obtain a fractional relaxation x_e for each edge e, and then have each right-node independently select an edge with probability x_e. However, this may cause the left-nodes to become unevenly loaded, leading to suboptimal solutions for some problems.
To address this, a number of algorithms for dependent rounding with strong negative correlation have been developed, e.g. Bansal, Srinivasan & Svensson (2021), Im & Shadloo (2020), Im & Li (2023), Harris (2024), Naor, Srinivasan & Wajc (2025). We introduce a new method for this, which we call the Dirichlet mechanism. It is based on having each left-node draw Dirichlet random variables for its edges, and then having each right-node select an edge based on these values. This achieves quantitatively stronger negative correlation than previous algorithms, and is also simpler since it avoids the need for a tie-breaking mechanism.
We illustrate the mechanism with improved approximation ratios for two problems. For oblivious online dependent rounding, we achieve a 0.68-approximation which improves upon the previous 0.652-approximation of Naor, Srinivasan & Wajc (2025). For the problem of scheduling jobs on unrelated machines to minimize weighted completion time, we achieve a 1.387-approximation which improves upon the 1.398-approximation of Harris (2024). (A recent algorithm of Li (2025) based on iterated rounding also provides a 1.36-approximation if the weights of each job are independent of machine.)