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Releasing All-Pairs Shortest Distances of Public Graphs with Differential Privacy

Tofighi Mohammadi, Alireza | 2023

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 56605 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Ebrahimi Boroojeni, Javad
  7. Abstract:
  8. In the context of differential privacy, a data holder has confidential information about users. The goal is to provide a randomized algorithm that takes this information as input and outputs an aggregation of the input. The algorithm must have the property that for any neighboring input pairs, the output distribution of the algorithm is close. One problem in differential privacy research is the release of shortest distances in a weighted graph. This model, first studied by Adam Sealfon, involves an edge-weighted graph G=(V, E) with weights w : E → R, where the topology of the graph is public and the private information is the weight of the edges. The aim is to provide an (ϵ, δ)-DP algorithm that takes as input w : E → R+ and releases an approximation of every shortest distance between every pair of vertices in G. In this definition, we say that inputs w, w ′ are neighbors if ∥w-w^' ∥_1≤1, and we define the error of the algorithm as the l_∞ distance between the result and the real distances. For general graphs, Chen et al. provided a lower bound of Ω(n^(1/6) ) for the error of a (ϵ,δ)-DP algorithm, where n is the number of vertices in the graph. They also provided an upper bound of O(n^(1/2) poly log⁡(n) log⁡(1/δ)  \/ϵ) for the error of an (ϵ,δ)-DP algorithm for this problem. Sealfon showed that it is possible to solve this problem for specific graphs, such as trees, with an error of O(log^2.5⁡(n)\/ϵ) with ϵ-DP guarantees. Fan et al. generalized this result for graphs with low feedback vertex set number. In this research, we reviewed and studied previous work and conducted a literature review. We then attempted to generalize their solutions and proofs and combine them into a single framework. We tried to solve this problem for graphs with small balanced vertex separators. We proved that if a graph and its subgraphs have balanced vertex separators with size k, we have a polynomial-time ϵ-DP algorithm that w.h.p has an error of at most O(k^2 poly log⁡(n)\/ϵ) and a polynomial-time (ϵ,δ)-DP algorithm that w.h.p has an error of at most O(k poly log⁡(n) log^0.5⁡(1\/δ)\/ϵ). Our algorithm can be considered as a generalization of Sealfon’s algorithm for trees.
  9. Keywords:
  10. Differential Privacy ; Randomized Algorithm ; Shortest Path ; Tree Width ; Tree Decomposition ; Graph Separators

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