We study the complexity of the problem of verifying differential privacy for whilelike programs working over boolean values and making probabilistic choices. Programs in this class can be interpreted into finite-state discrete-time Markov Chains (DTMC). We show that the problem of deciding whether a program is differentially private for specific values of the privacy parameters is PSPACE-complete. To show that this problem is in PSPACE, we adapt classical results about computing hitting probabilities for DTMC. To show PSPACE-hardness we use a reduction from the problem of checking whether a program almost surely terminates or not. We also show that the problem of approximating the privacy parameters that a program provides is PSPACE-hard. Moreover, we investigate the complexity of similar problems also for several relaxations of differential privacy: Rényi differential privacy, concentrated differential privacy, and truncated concentrated differential privacy. For these notions, we consider gap-versions of the problem of deciding whether a program is private or not and we show that all of them are PSPACE-complete.

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We show that assuming the Exponential Time Hypothesis, the Partial Minimum Branching Program Size Problem (MBPSP * ) requires superpolynomial time. This result also applies to the partial minimization problems for many interesting subclasses of branching programs, such as read-k branching programs and OBDDs.Combining these results with our recent result (Glinskih and Riazanov, LATIN 2022) we obtain an unconditional superpolynomial lower bound on the size of Read-Once Nondeterministic Branching Programs (1-NBP) computing the total versions of the minimum BP, read-k-BP, and OBDD size problems.Additionally we show that it is NP-hard to check whether a given BP computing a partial Boolean function can be compressed to a BP of a given size.1 The same hardness result holds for Partial MFSP (MFSP * ).2 Suppose that f is such a reduction from MCSP s=3n (the language of the truth-tables computable by circuits with at most 3n gates) to MBPSP s=n 2.1 . Then taking any function requiring circuits of size larger than 3n [FGHK16, LY22] and applying f to it yields n 2.1 lower bound for general branching programs superior to the current state of the art. In a similar way, we get superlinear circuit lower bounds from any reduction from MBPSP s=n 2 to MCSP s=n 1+ε .3 We note that results in this area often establish NP-hardness and even hardness of size approximation for the case of concise representation, while our result only establishes ETH-hardness of the exact version of the problem. 4 The input to this problem has Θ(n 4 log n) bit size.