Differential Privacy Meets Fixed Parameter Tractability: Algorithms and Lower Bounds
Theory paper combines differential privacy with fixed-parameter tractable encoders, improving approximation guarantees for combinatorial optimization and proving new lower bounds.
The paper studies combinatorial optimization under epsilon-differential privacy within the implicit encoder-decoder framework of Gupta et al. (SODA 2010), generalizing it to allow fixed-parameter tractable encoders. This circumvents approximation barriers inherent to polynomial-time algorithms and yields improved guarantees for fundamental combinatorial optimization problems. The authors establish the first representation-independent lower bounds: assuming a non-uniform variant of the Gap Exponential Time Hypothesis, no epsilon-DP encoder-decoder pair can achieve certain approximation guarantees with a subexponential-time decoder for sufficiently small epsilon. Representation-dependent lower bounds are also provided for larger epsilon.
- Extends private encoder-decoder framework with fixed-parameter tractable encoders
- Circumvents polynomial-time approximation barriers for DP optimization
- First representation-independent lower bounds under non-uniform Gap ETH
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We study combinatorial optimization problems under the constraint of $ε$-differential privacy ($ε$-DP). Given the strong lower bounds for explicitly outputting solutions, we work within the implicit representation framework of Gupta et al. (SODA 2010), where a private polynomial-time randomized "encoder" generates a representation of a solution, and a "decoder" uses this representation along with the input to extract a valid final solution. In this work, we generalize this framework by allowing the encoder to run in fixed-parameter tractable time. This circumvents approximation barriers inherent to polynomial-time algorithms and obtains improved guarantees for many fundamental combinatorial optimization problems. Finally, we establish the first representation-independent lower bounds for our framework. Assuming a non-uniform variant of the Gap Exponential Time Hypothesis, for sufficiently small $ε> 0$, we prove that no $ε$-DP encoder-decoder pair can achieve certain approximation guarantees, if the decoder runs in subexponential time. We further provide representation-dependent lower bounds that hold even for larger $ε$.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.12508