complexity theorists

Abstract: 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 $ε$.

Abstract: We define and study a measure of information complexity for randomized decision trees. We prove three main results about this complexity measure: Information equals amortized size complexity. We show that the information complexity of randomized decision tree is equal to the logarithm of the amortized worst-case randomized tree size complexity of computing a function f. That is, when computing f on n inputs, the logarithm of the randomized tree size is exactly equal to the amount of information needed to compute the function. Information allows for tree size compression. We show that even when computing f on a single input, the information complexity can be used to compress the size of a tree, if we allow a small loss in success probability. With the recent characterization of Chattopadhyay, Dahiya, Mande, Radhakrishnan, and Sanyal (2023), this result shows that the depth of AND-OR trees can also be compressed in terms of information complexity. Direct Product Theorems. We show that the success-conditioned variant of information complexity satisfies a perfect direct product theorem. This result gives an information complexity analogue of the direct product theorem for success-conditioned randomized query complexity by Ben-David and Blais (2025).

Abstract: This project began as an attempt to prove that Medvedev logic is decidable with the help of generative AI systems. The author (as well as the generative AI systems, or at least they claim to be since I have asked them) was surprised by its eventual conclusion. We prove that Medvedev logic ML, the intermediate logic of finite problems, is Pi-01-complete under computable many-one reductions. Consequently, ML is not recursively enumerable, a fortiori undecidable, and admits no recursively enumerable sound and complete proof calculus. The proof connects the periodic domino problem with intuitionistic formulas through a shared intermediate structure that we call a Wang-Medvedev pair. Such a pair consists of a finite partially ordered set of roles together with demands. Demands define the interaction between roles. A realization labels nonempty subsets of a finite set with these roles, respecting the order and satisfying the demands. We associate a pair with each finite Wang system and show that it has a realization iff the system tiles a finite torus. We then construct an intuitionistic formula that fails on some finite Medvedev frame iff the same pair is realizable. Realizability thus provides the link between periodic tilings and the countermodels.

Abstract: Relativization is concerned with comparing computational models with black-box access to an oracle. For promise problems, black-box access is not canonical due to inputs outside of the promise being unconstrained. We study two semantics for such access. Under robust queries, a machine must correctly answer regardless of the completion of the problem,, while loose access requires that the internal choices of a machine do not change based on off-promise queries. Our first result separates the language and promise settings. Namely, we construct an oracle $O$ such that $\mathsf{P}^O = \mathsf{BQP}^O = \mathsf{AWPP}^O$, but $\mathsf{PromiseBQP}^O\not\subseteq\mathsf{PromiseP}^O_{\mathsf{/poly}}$. In particular, $\mathsf{BPP}^O = \mathsf{BQP}^O$, but $\mathsf{PromiseBQP}^O \neq \mathsf{PromiseBPP}^O$, showing that results for languages need not transfer to promises. Next, we use loose queries to strengthen the upper bound on the Quantum-Classical Polynomial Hierarchy from $\mathsf{P}^{\mathsf{PP}^{\mathsf{PP}}}$ to $\mathsf{QCPH} \subseteq \mathsf{BP\cdot PP} \subseteq \mathsf{PromiseBPP}^{\mathsf{PP}}$. The same proof also shows $\mathsf{PP}^\mathsf{PromiseBQP} = \mathsf{PP}$. Additionally, we show that $\mathsf{PromiseBQP}$, even when given quantum advice, is self-low under robust queries. Finally, we exhibit an obstruction to transferring language-level counting results to promise classes. Although $\mathsf{AWPP}$ and $\mathsf{APP}$ are low for $\mathsf{PP}$, a corresponding promise analogue would collapse the counting hierarchy as $\mathsf{GapP} \subseteq \mathsf{FP}^{\mathsf{PromiseAWPP}}$. This motivates the introduction of $\mathsf{PromisePostBQP^*}$, which restricts $\mathsf{PostBQP}$ to input-indepencent postselection. By showing that it is low for \PP, we obtain $\mathsf{PP}^{\mathsf{PromiseYQP^*}} = \mathsf{PP}$.

Abstract: We show that 2-fold Forrelation with inverse-polylogarithmic promise gap can be solved, with bounded error, by polynomial-size QAC$^0$ circuits. Unlike the standard oracle-based Forrelation algorithm, our circuits receive the input explicitly, in the same form as the AC$^0$ circuits against which Forrelation is known to be hard. At constant gap, this yields a natural promise-problem separation between QAC$^0$ and AC$^0$.