Optimal Lower Bounds for Universal and Differentially Private Steiner Trees and TSPs

Given a metric space on n points, an α-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 α factor of the optimal Steiner tree cost for X. An α-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 α 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 α-approximate universal algorithm for the Steiner tree problem or TSP is also an α-approximate differentially private algorithm. It is known that both problems admit O(log n)-approximate universal algorithms, and hence O(log n)-approximate differentially private algorithms as well. We prove an Ω(log n) 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. This improves upon an earlier Ω(log n/ log log n) lower bound for the universal Steiner tree problem, and an Ω(log n) lower bound for the universal TSP. The latter answers an open question in Hajiaghayi et al. [13]. When expressed as a function of the size of the input subset of vertices, say k, our lower bounds are in fact Ω(k) for both problems, improving upon the previously known log k lower bounds. We then show that whenever the universal problem has a lower bound that satisfies an additional property, it implies a similar lower bound for the differentially private version. Using this converse relation between universal and private algorithms, we establish an Ω(log n) lower bound for the differentially private Steiner tree and the differentially private TSP. This answers a question of Talwar [27]. Our results highlight a natural connection between universal and private approximation algorithms that is likely to have other applications. ∗Department of Computer and Information Science, University of Pennsylvania, Philadelphia PA. Email: [email protected]. Supported by NSF Award CCF-0635084 and IIS-0904314. †Department of Computer and Information Science, University of Pennsylvania, Philadelphia PA. Email: [email protected]. ‡Dept. of Computer & Information Science, University of Pennsylvania, Philadelphia, PA 19104. Email: [email protected]. Supported in part by NSF Awards CCF-0635084 and IIS-0904314.