# Optimal Lower Bounds for Universal and Differentially Private Steiner Tree and TSP - Computer Science > Data Structures and Algorithms

Optimal Lower Bounds for Universal and Differentially Private Steiner Tree and TSP - Computer Science > Data Structures and Algorithms - Download this document for free, or read online. Document in PDF available to download.

Abstract: Given a metric space on n points, an {\alpha}-approximate universal algorithmfor the Steiner tree problem outputs a distribution over rooted spanning treessuch that for any subset X of vertices containing the root, the expected costof the induced subtree is within an {\alpha} factor of the optimal Steiner treecost for X. An {\alpha}-approximate differentially private algorithm for theSteiner tree problem takes as input a subset X of vertices, and outputs a treedistribution that induces a solution within an {\alpha} factor of the optimalas before, and satisfies the additional property that for any set X- thatdiffers in a single vertex from X, the tree distributions for X and X- are-close- to each other. Universal and differentially private algorithms for TSPare defined similarly. An {\alpha}-approximate universal algorithm for theSteiner tree problem or TSP is also an {\alpha}-approximate differentiallyprivate algorithm. It is known that both problems admit Ologn-approximateuniversal algorithms, and hence Olog n-approximate differentially privatealgorithms as well. We prove an {\Omega}logn lower bound on the approximationratio achievable for the universal Steiner tree problem and the universal TSP,matching the known upper bounds. Our lower bound for the Steiner tree problemholds even when the algorithm is allowed to output a more general solution of adistribution on paths to the root.

Author: ** Anand Bhalgat, Deeparnab Chakrabarty, Sanjeev Khanna**

Source: https://arxiv.org/