On approximating arbitrary metrices by tree metrics
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
An O(log*n) approximation algorithm for the asymmetric p-center problem
Journal of Algorithms
Two O (log* k)-Approximation Algorithms for the Asymmetric k-Center Problem
Proceedings of the 8th International IPCO Conference on Integer Programming and Combinatorial Optimization
A tight bound on approximating arbitrary metrics by tree metrics
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Universal approximations for TSP, Steiner tree, and set cover
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
Improved lower and upper bounds for universal TSP in planar metrics
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Probabilistic computations: Toward a unified measure of complexity
SFCS '77 Proceedings of the 18th Annual Symposium on Foundations of Computer Science
SFCS '88 Proceedings of the 29th Annual Symposium on Foundations of Computer Science
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Extensions and limits to vertex sparsification
Proceedings of the forty-second ACM symposium on Theory of computing
Differentially private combinatorial optimization
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Improved lower bounds for the universal and a priori TSP
APPROX/RANDOM'10 Proceedings of the 13th international conference on Approximation, and 14 the International conference on Randomization, and combinatorial optimization: algorithms and techniques
ICALP'06 Proceedings of the 33rd international conference on Automata, Languages and Programming - Volume Part II
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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 a 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 a 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. 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 [19]. Our results highlight a natural connection between universal and private approximation algorithms that is likely to have other applications.