🤖 AI Summary
This work addresses the issue of load imbalance on left nodes and degraded solution quality caused by independent randomized rounding in assignment graphs. To overcome this, the authors propose a dependent rounding mechanism based on the Dirichlet distribution: each left node generates Dirichlet-distributed random variables over its incident edges, and right nodes select edges accordingly, naturally inducing strong negative correlations. The method is structurally simple, avoids complex tie-breaking strategies, and achieves stronger negative correlation than existing approaches. It yields improved approximation guarantees—0.68 for oblivious online dependent rounding (surpassing the previous 0.652) and 1.387 for unrelated-machine scheduling to minimize weighted completion time (improving upon the prior 1.398)—demonstrating significantly enhanced rounding performance for fractional solutions in combinatorial optimization.
📝 Abstract
Many optimization and scheduling problems can be abstracted in terms of a bipartite ``assignment graph" $G = (L \cup R, E)$, 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 \emph{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.)