# Complexity of Counting

Murtagh, Jack, and Salil Vadhan. “The complexity of computing the optimal composition of differential privacy.” Theory of Computing 14 (2018): 1-35. Publisher's VersionAbstract

Version History: Full version posted on CoRR, abs/1507.03113, July 2015Additional version published in Proceedings of the 13th IACR Theory of Cryptography Conference (TCC '16-A)

In the study of differential privacy, composition theorems (starting with the original paper of Dwork, McSherry, Nissim, and Smith (TCC '06)) bound the degradation of privacy when composing several differentially private algorithms. Kairouz, Oh, and Viswanath (ICML '15) showed how to compute the optimal bound for composing $$k$$ arbitrary ($$\epsilon$$,$$\delta$$)- differentially private algorithms. We characterize the optimal composition for the more general case of $$k$$ arbitrary ($$\epsilon_1$$ , $$\delta_1$$ ), . . . , ($$\epsilon_k$$ , $$\delta_k$$ )-differentially private algorithms where the privacy parameters may differ for each algorithm in the composition. We show that computing the optimal composition in general is $$\#$$P-complete. Since computing optimal composition exactly is infeasible (unless FP$$=$$$$\#$$P), we give an approximation algorithm that computes the composition to arbitrary accuracy in polynomial time. The algorithm is a modification of Dyer’s dynamic programming approach to approximately counting solutions to knapsack problems (STOC '03).

Thaler, Justin, Jonathan Ullman, and Salil Vadhan. “Faster algorithms for privately releasing marginals.” In Artur Czumaj, Kurt Mehlhorn, Andrew M. Pitts, and Roger Wattenhofer, editors, Proceedings of the 39th International Colloquium on Automata, Languages, and Programming (ICALP ‘12), Lecture Notes on Computer Science, 7391:810-821. Springer-Verlag, 2012. Publisher's VersionAbstract

Version History: Full version posted as arXiv:1205.1758v2.

We study the problem of releasing k-way marginals of a database $$D ∈ ({0,1}^d)^n$$, while preserving differential privacy. The answer to a $$k$$-way marginal query is the fraction of $$D$$’s records $$x \in \{0,1\}^d$$ with a given value in each of a given set of up to $$k$$ columns. Marginal queries enable a rich class of statistical analyses of a dataset, and designing efficient algorithms for privately releasing marginal queries has been identified as an important open problem in private data analysis (cf. Barak et. al., PODS ’07).

We give an algorithm that runs in time $$d^{O(\sqrt{K})}$$ and releases a private summary capable of answering any $$k$$-way marginal query with at most ±.01 error on every query as long as $$n \geq d^{O(\sqrt{K})}$$. To our knowledge, ours is the first algorithm capable of privately releasing marginal queries with non-trivial worst-case accuracy guarantees in time substantially smaller than the number of k-way marginal queries, which is $$d^{\Theta(k)}$$ (for $$k ≪ d$$).

Vadhan, Salil. “The complexity of counting in sparse, regular, and planar graphs.SIAM Journal on Computing 31, no. 2 (2001): 398-427.Abstract
We show that a number of graph-theoretic counting problems remain NP-hard, indeed #P-complete, in very restricted classes of graphs. In particular, we prove that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree. We obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2-CNF formulae. To achieve these results, a new interpolation-based reduction technique which preserves properties such as constant degree is introduced.