George Z. Li

George Z. Li, Dung Nguyen, Anil Vullikanti

It was observed in \citet{gupta2009differentially} that the Set Cover problem has strong impossibility results under differential privacy. In our work, we observe that these hardness results dissolve when we turn to the Partial Set Cover problem, where we only need to cover a $ρ$-fraction of the elements in the universe, for some $ρ\in(0,1)$. We show that this relaxation enables us to avoid the impossibility results: under loose conditions on the input set system, we give differentially private algorithms which output an explicit set cover with non-trivial approximation guarantees. In particular, this is the first differentially private algorithm which outputs an explicit set cover. Using our algorithm for Partial Set Cover as a subroutine, we give a differentially private (bicriteria) approximation algorithm for a facility location problem which generalizes $k$-center/$k$-supplier with outliers. Like with the Set Cover problem, no algorithm has been able to give non-trivial guarantees for $k$-center/$k$-supplier-type facility location problems due to the high sensitivity and impossibility results. Our algorithm shows that relaxing the covering requirement to serving only a $ρ$-fraction of the population, for $ρ\in(0,1)$, enables us to circumvent the inherent hardness. Overall, our work is an important step in tackling and understanding impossibility results in private combinatorial optimization.

David G. Harris, George Z. Li, Nitya Raju et al.

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.)