Optimal Lower Bounds for Universal and Differentially Private Steiner Tree and TSP

Given a metric space on n points, an {\alpha}-approximate universal algorithm for the Steiner tree problem outputs a distribution over rooted spanning trees such that for any subset X of vertices containing the root, the expected cost of the induced subtree is within an {\alpha} factor of the optimal Steiner tree cost for X. An {\alpha}-approximate differentially private algorithm for the Steiner tree problem takes as input a subset X of vertices, and outputs a tree distribution that induces a solution within an {\alpha} factor of the optimal as before, and satisfies the additional property that for any set X' that differs in a single vertex from X, the tree distributions for X and X' are "close" to each other. Universal and differentially private algorithms for TSP are defined similarly. An {\alpha}-approximate universal algorithm for the Steiner tree problem or TSP is also an {\alpha}-approximate differentially private algorithm. It is known that both problems admit O(logn)-approximate universal algorithms, and hence O(log n)-approximate differentially private algorithms as well. We prove an {\Omega}(logn) lower bound on the approximation ratio achievable for the universal Steiner tree problem and the universal TSP, matching the known upper bounds. Our lower bound for the Steiner tree problem holds even when the algorithm is allowed to output a more general solution of a distribution on paths to the root.Comment: 14 page

Differential Privacy from Locally Adjustable Graph Algorithms: kk-Core Decomposition, Low Out-Degree Ordering, and Densest Subgraphs

Differentially private algorithms allow large-scale data analytics while preserving user privacy. Designing such algorithms for graph data is gaining importance with the growth of large networks that model various (sensitive) relationships between individuals. While there exists a rich history of important literature in this space, to the best of our knowledge, no results formalize a relationship between certain parallel and distributed graph algorithms and differentially private graph analysis. In this paper, we define \emph{locally adjustable} graph algorithms and show that algorithms of this type can be transformed into differentially private algorithms. Our formalization is motivated by a set of results that we present in the central and local models of differential privacy for a number of problems, including $k$-core decomposition, low out-degree ordering, and densest subgraphs. First, we design an $\varepsilon$-edge differentially private (DP) algorithm that returns a subset of nodes that induce a subgraph of density at least $\frac{D^*}{1+\eta} - O\left(\text{poly}(\log n)/\varepsilon\right),$ where $D^*$ is the density of the densest subgraph in the input graph (for any constant $\eta > 0$). This algorithm achieves a two-fold improvement on the multiplicative approximation factor of the previously best-known private densest subgraph algorithms while maintaining a near-linear runtime. Then, we present an $\varepsilon$-locally edge differentially private (LEDP) algorithm for $k$-core decompositions. Our LEDP algorithm provides approximates the core numbers (for any constant $\eta > 0$) with $(2+\eta)$ multiplicative and $O\left(\text{poly}\left(\log n\right)/\varepsilon\right)$ additive error. This is the first differentially private algorithm that outputs private $k$-core decomposition statistics