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

Differentially Private Partial Set Cover with Applications to Facility Location

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 $\rho$-fraction of the elements in the universe, for some $\rho\in(0,1)$. 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 $\rho$-fraction of the population, for $\rho\in(0,1)$, enables us to circumvent the inherent hardness. Overall, our work is an important step in tackling and understanding impossibility results in private combinatorial optimization.Comment: 11 pages, 2 figures. Full version of IJCAI 2023 publicatio