Differentially Private Algorithms for Graph Cuts: A Shifting Mechanism Approach and More

• URL: http://arxiv.org/abs/2407.06911v4
• Date: Fri, 08 Nov 2024 02:03:48 GMT
• Title: Differentially Private Algorithms for Graph Cuts: A Shifting Mechanism Approach and More
• Authors: Rishi Chandra, Michael Dinitz, Chenglin Fan, Zongrui Zou,
• Abstract summary: We introduce edgedifferentially private algorithms for the multiway cut and the minimum $k$cut. For the minimum $k$-cut problem we use a different approach, combining the exponential mechanism with bounds on the number of approximate $k$-cuts.
• Score: 5.893651469750359
• License:
• Abstract: In this paper, we address the challenge of differential privacy in the context of graph cuts, specifically focusing on the multiway cut and the minimum $k$-cut. We introduce edge-differentially private algorithms that achieve nearly optimal performance for these problems. Motivated by multiway cut, we propose the shifting mechanism, a general framework for private combinatorial optimization problems. This framework allows us to develop an efficient private algorithm with a multiplicative approximation ratio that matches the state-of-the-art non-private algorithm, improving over previous private algorithms that have provably worse multiplicative loss. We then provide a tight information-theoretic lower bound on the additive error, demonstrating that for constant $k$, our algorithm is optimal in terms of the privacy cost. The shifting mechanism also allows us to design private algorithm for the multicut and max-cut problems, with runtimes determined by the best non-private algorithms for these tasks. For the minimum $k$-cut problem we use a different approach, combining the exponential mechanism with bounds on the number of approximate $k$-cuts to get the first private algorithm with optimal additive error of $O(k\log n)$ (for a fixed privacy parameter). We also establish an information-theoretic lower bound that matches this additive error. Furthermore, we provide an efficient private algorithm even for non-constant $k$, including a polynomial-time 2-approximation with an additive error of $\tilde{O}(k^{1.5})$.

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