arXiv subjects

Zeyong Li

Publications and source records attributed to Zeyong Li.

10 recordsLinked to original sources

arXiv2025

Hierarchies within TFNP: building blocks and collapses

In all well-studied $\mathsf{TFNP}$ subclasses (e.g. $\mathsf{PPA}, \mathsf{PPP}$ etc.), the canonical complete problem takes as input a polynomial-size circuit $C: \{ 0, 1\}^n \rightarrow \{ 0, 1\}^m$ whose input-output behavior implicitly encodes an exponentially large object $G$, i.e. $C$ is the succinct (polynomial-size) representation of the exponential size object $G$. The goal is to find some particular substructure in $G$ which can be confirmed in polynomial time using queries to $C$. We initiate the study of classes of the form $\mathsf{A}^{\mathsf{B}}$ where both $\mathsf{A}$ and $\mathsf{B}$ are $\mathsf{TFNP}$ subclasses. In particular, we define complete problems for these classes that take as input a circuit $C$ which is allowed oracle gates to another $\mathsf{TFNP}$ class. Beyond introducing definitions for $\mathsf{TFNP}$ oracle problems, our specific technical contributions include showing that several $\mathsf{TFNP}$ subclasses are self-low and hence their corresponding hierarchies collapse. In particular, $\mathsf{PPA^{PPA}} = \mathsf{PPA}$, $\mathsf{PLS^{PLS}} = \mathsf{PLS}$, and $\mathsf{LOSSY^{LOSSY}} = \mathsf{LOSSY}$. As an immediate consequence, we derive that when reducing to $\mathsf{PPA}$, one can always assume access to $\mathsf{PPA}$ -- and therefore factoring -- oracle gates. In addition to introducing a variety of hierarchies within $\mathsf{TFNP}$ that merit study in their own right, these ideas introduce a novel approach for classifying computational problems within $\mathsf{TFNP}$ and proving black-box separations. For example, we observe that the problem of deterministically generating large prime numbers, which has long resisted classification in a $\mathsf{TFNP}$ subclass, is in $\mathsf{PPP^{\mathsf{PPP}}}$ under the Generalized Riemann Hypothesis.

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arXiv2025

Oblivious Complexity Classes Revisited: Lower Bounds and Hierarchies

In this work we study oblivious complexity classes. These classes capture the power of interactive proofs where the prover(s) are only given the input size rather than the actual input. In particular, we study the connections between the symmetric polynomial time $\mathsf{S_2P}$ and its oblivious counterpart $\mathsf{O_2P}$. Among our results, we construct an explicit language in $\mathsf{O_2P}$ that cannot be computed by circuits of size $n^k$, and thus prove a hierarchy theorem for $\mathsf{O_2TIME}$. Along the way we also make partial progress towards the resolution of an open question posed by Goldreich and Meir (TOCT 2015) that relates the complexity of $\mathsf{NP}$ to its oblivious counterpart $\mathsf{ONP}$. To the best of our knowledge, these results constitute the first explicit fixed-polynomial lower bound and hierarchy theorem for $\mathsf{O_2P}$. The smallest uniform complexity class for which such lower bounds were previously known was $\mathsf{S_2P}$, due to Cai (JCSS 2007). In addition, this is the first uniform hierarchy theorem for a semantic class. All previous results required some non-uniformity.

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arXiv2025

Downward self-reducibility in the total function polynomial hierarchy

A problem $\mathcal{P}$ is considered downward self-reducible, if there exists an efficient algorithm for $\mathcal{P}$ that is allowed to make queries to only strictly smaller instances of $\mathcal{P}$. Downward self-reducibility has been well studied in the case of decision problems, and it is well known that any downward self-reducible problem must lie in $\mathsf{PSPACE}$. Harsha, Mitropolsky and Rosen [ITCS, 2023] initiated the study of downward self reductions in the case of search problems. They showed the following interesting collapse: if a problem is in $\mathsf{TFNP}$ and also downward self-reducible, then it must be in $\mathsf{PLS}$. Moreover, if the problem admits a unique solution then it must be in $\mathsf{UEOPL}$. We demonstrate that this represents just the tip of a much more general phenomenon, which holds for even harder search problems that lie higher up in the total function polynomial hierarchy ($\mathsf{TFΣ_i^P}$). In fact, even if we allow our downward self-reduction to be much more powerful, such a collapse will still occur. We show that any problem in $\mathsf{TFΣ_i^P}$ which admits a randomized downward self-reduction with access to a $\mathsf{Σ_{i-1}^P}$ oracle must be in $\mathsf{PLS}^{\mathsf{Σ_{i-1}^P}}$. If the problem has \textit{essentially unique solutions} then it lies in $\mathsf{UEOPL}^{\mathsf{Σ_{i-1}^P}}$. As one (out of many) application of our framework, we get new upper bounds for the problems $\mathrm{Range Avoidance}$ and $\mathrm{Linear Ordering Principle}$ and show that they are both in $\mathsf{UEOPL}^{\mathsf{NP}}$.

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arXiv2024

Symmetric Exponential Time Requires Near-Maximum Circuit Size: Simplified, Truly Uniform

In a recent breakthrough, Chen, Hirahara and Ren prove that $\mathsf{S_2E}/_1 \not\subset \mathsf{SIZE}[2^n/n]$ by giving a single-valued $\mathsf{FS_2P}$ algorithm for the Range Avoidance Problem ($\mathsf{Avoid}$) that works for infinitely many input size $n$. Building on their work, we present a simple single-valued $\mathsf{FS_2P}$ algorithm for $\mathsf{Avoid}$ that works for all input size $n$. As a result, we obtain the circuit lower bound $\mathsf{S_2E} \not\subset {i.o.}$-$\mathsf{SIZE}[2^n/n]$ and many other corollaries: 1. Almost-everywhere near-maximum circuit lower bound for $\mathsf{Σ_2E} \cap \mathsf{Π_2E}$ and $\mathsf{ZPE}^{\mathsf{NP}}$. 2. Pseudodeterministic $\mathsf{FZPP}^{\mathsf{NP}}$ constructions for: Ramsey graphs, rigid matrices, pseudorandom generators, two-source extractors, linear codes, hard truth tables, and $K^{poly}$-random strings.

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arXiv2023

Separating Key Agreement and Computational Differential Privacy

Two party differential privacy allows two parties who do not trust each other, to come together and perform a joint analysis on their data whilst maintaining individual-level privacy. We show that any efficient, computationally differentially private protocol that has black-box access to key agreement (and nothing stronger), is also an efficient, information-theoretically differentially private protocol. In other words, the existence of efficient key agreement protocols is insufficient for efficient, computationally differentially private protocols. In doing so, we make progress in answering an open question posed by Vadhan about the minimal computational assumption needed for computational differential privacy. Combined with the information-theoretic lower bound due to McGregor, Mironov, Pitassi, Reingold, Talwar, and Vadhan in [FOCS'10], we show that there is no fully black-box reduction from efficient, computationally differentially private protocols for computing the Hamming distance (or equivalently inner product over the integers) on $n$ bits, with additive error lower than $O\left(\frac{\sqrt{n}}{e^ε\log(n)}\right)$, to key agreement. This complements the result by Haitner, Mazor, Silbak, and Tsfadia in [STOC'22], which showed that computing the Hamming distance implies key agreement. We conclude that key agreement is \emph{strictly} weaker than computational differential privacy for computing the inner product, thereby answering their open question on whether key agreement is sufficient.

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arXiv2023

The Complexity of Distributed Approximation of Packing and Covering Integer Linear Programs

In this paper, we present a low-diameter decomposition algorithm in the LOCAL model of distributed computing that succeeds with probability $1 - 1/poly(n)$. Specifically, we show how to compute an $\left(ε, O\left(\frac{\log n}ε\right)\right)$ low-diameter decomposition in $O\left(\frac{\log^3(1/ε)\log n}ε\right)$ round Further developing our techniques, we show new distributed algorithms for approximating general packing and covering integer linear programs in the LOCAL model. For packing problems, our algorithm finds an $(1-ε)$-approximate solution in $O\left(\frac{\log^3 (1/ε) \log n}ε\right)$ rounds with probability $1 - 1/poly(n)$. For covering problems, our algorithm finds an $(1+ε)$-approximate solution in $O\left(\frac{\left(\log \log n + \log (1/ε)\right)^3 \log n}ε\right)$ rounds with probability $1 - 1/poly(n)$. These results improve upon the previous $O\left(\frac{\log^3 n}ε\right)$-round algorithm by Ghaffari, Kuhn, and Maus [STOC 2017] which is based on network decompositions. Our algorithms are near-optimal for many fundamental combinatorial graph optimization problems in the LOCAL model, such as minimum vertex cover and minimum dominating set, as their $(1\pm ε)$-approximate solutions require $Ω\left(\frac{\log n}ε\right)$ rounds to compute.

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arXiv2022

Lattice Problems Beyond Polynomial Time

We study the complexity of lattice problems in a world where algorithms, reductions, and protocols can run in superpolynomial time, revisiting four foundational results: two worst-case to average-case reductions and two protocols. We also show a novel protocol. 1. We prove that secret-key cryptography exists if $\widetilde{O}(\sqrt{n})$-approximate SVP is hard for $2^{\varepsilon n}$-time algorithms. I.e., we extend to our setting (Micciancio and Regev's improved version of) Ajtai's celebrated polynomial-time worst-case to average-case reduction from $\widetilde{O}(n)$-approximate SVP to SIS. 2. We prove that public-key cryptography exists if $\widetilde{O}(n)$-approximate SVP is hard for $2^{\varepsilon n}$-time algorithms. This extends to our setting Regev's celebrated polynomial-time worst-case to average-case reduction from $\widetilde{O}(n^{1.5})$-approximate SVP to LWE. In fact, Regev's reduction is quantum, but ours is classical, generalizing Peikert's polynomial-time classical reduction from $\widetilde{O}(n^2)$-approximate SVP. 3. We show a $2^{\varepsilon n}$-time coAM protocol for $O(1)$-approximate CVP, generalizing the celebrated polynomial-time protocol for $O(\sqrt{n/\log n})$-CVP due to Goldreich and Goldwasser. These results show complexity-theoretic barriers to extending the recent line of fine-grained hardness results for CVP and SVP to larger approximation factors. (This result also extends to arbitrary norms.) 4. We show a $2^{\varepsilon n}$-time co-non-deterministic protocol for $O(\sqrt{\log n})$-approximate SVP, generalizing the (also celebrated!) polynomial-time protocol for $O(\sqrt{n})$-CVP due to Aharonov and Regev. 5. We give a novel coMA protocol for $O(1)$-approximate CVP with a $2^{\varepsilon n}$-time verifier. All of the results described above are special cases of more general theorems that achieve time-approximation factor tradeoffs.

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arXiv2022

Alternating Automatic Register Machines

This paper introduces and studies a new model of computation called an Alternating Automatic Register Machine (AARM). An AARM possesses the basic features of a conventional register machine and an alternating Turing machine, but can carry out computations using bounded automatic relations in a single step. One finding is that an AARM can recognise some NP-complete problems, including SAT (using a particular coding), in $\log^* n + O(1)$ steps. On the other hand, if all problems in P can be solved by an AARM in $O(\log^*n)$ rounds, then $\text{P} \subset \text{PSPACE}$. Furthermore, we study an even more computationally powerful machine, called a Polynomial-Size Padded Alternating Automatic Register Machine (PAARM), which allows the input to be padded with a polynomial-size string. It is shown that the polynomial hierarchy can be characterised as the languages that are recognised by a PAARM in $\log^*n + O(1)$ steps. These results illustrate the power of alternation when combined with computations involving automatic relations, and uncover a finer gradation between known complexity classes.

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arXiv2021

Dimension-Preserving Reductions Between SVP and CVP in Different $p$-Norms

$ \newcommand{\SVP}{\textsf{SVP}} \newcommand{\CVP}{\textsf{CVP}} \newcommand{\eps}{\varepsilon} $We show a number of reductions between the Shortest Vector Problem and the Closest Vector Problem over lattices in different $\ell_p$ norms ($\SVP_p$ and $\CVP_p$ respectively). Specifically, we present the following $2^{\eps m}$-time reductions for $1 \leq p \leq q \leq \infty$, which all increase the rank $n$ and dimension $m$ of the input lattice by at most one: $\bullet$ a reduction from $\widetilde{O}(1/\eps^{1/p})γ$-approximate $\SVP_q$ to $γ$-approximate $\SVP_p$; $\bullet$ a reduction from $\widetilde{O}(1/\eps^{1/p}) γ$-approximate $\CVP_p$ to $γ$-approximate $\CVP_q$; and $\bullet$ a reduction from $\widetilde{O}(1/\eps^{1+1/p})$-$\CVP_q$ to $(1+\eps)$-unique $\SVP_p$ (which in turn trivially reduces to $(1+\eps)$-approximate $\SVP_p$). The last reduction is interesting even in the case $p = q$. In particular, this special case subsumes much prior work adapting $2^{O(m)}$-time $\SVP_p$ algorithms to solve $O(1)$-approximate $\CVP_p$. In the (important) special case when $p = q$, $1 \leq p \leq 2$, and the $\SVP_p$ oracle is exact, we show a stronger reduction, from $O(1/\eps^{1/p})\text{-}\CVP_p$ to (exact) $\SVP_p$ in $2^{\eps m}$ time. For example, taking $\eps = \log m/m$ and $p = 2$ gives a slight improvement over Kannan's celebrated polynomial-time reduction from $\sqrt{m}\text{-}\CVP_2$ to $\SVP_2$. We also note that the last two reductions can be combined to give a reduction from approximate-$\CVP_p$ to $\SVP_q$ for any $p$ and $q$, regardless of whether $p \leq q$ or $p > q$. Our techniques combine those from the recent breakthrough work of Eisenbrand and Venzin (which showed how to adapt the current fastest known algorithm for these problems in the $\ell_2$ norm to all $\ell_p$ norms) together with sparsification-based techniques.

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arXiv2020

A $2^{n/2}$-Time Algorithm for $\sqrt{n}$-SVP and $\sqrt{n}$-Hermite SVP, and an Improved Time-Approximation Tradeoff for (H)SVP

We show a $2^{n/2+o(n)}$-time algorithm that finds a (non-zero) vector in a lattice $\mathcal{L} \subset \mathbb{R}^n$ with norm at most $\tilde{O}(\sqrt{n})\cdot \min\{λ_1(\mathcal{L}), \det(\mathcal{L})^{1/n}\}$, where $λ_1(\mathcal{L})$ is the length of a shortest non-zero lattice vector and $\det(\mathcal{L})$ is the lattice determinant. Minkowski showed that $λ_1(\mathcal{L}) \leq \sqrt{n} \det(\mathcal{L})^{1/n}$ and that there exist lattices with $λ_1(\mathcal{L}) \geq Ω(\sqrt{n}) \cdot \det(\mathcal{L})^{1/n}$, so that our algorithm finds vectors that are as short as possible relative to the determinant (up to a polylogarithmic factor). The main technical contribution behind this result is new analysis of (a simpler variant of) an algorithm from arXiv:1412.7994, which was only previously known to solve less useful problems. To achieve this, we rely crucially on the ``reverse Minkowski theorem'' (conjectured by Dadush arXiv:1606.06913 and proven by arXiv:1611.05979), which can be thought of as a partial converse to the fact that $λ_1(\mathcal{L}) \leq \sqrt{n} \det(\mathcal{L})^{1/n}$. Previously, the fastest known algorithm for finding such a vector was the $2^{.802n + o(n)}$-time algorithm due to [Liu, Wang, Xu, and Zheng, 2011], which actually found a non-zero lattice vector with length $O(1) \cdot λ_1(\mathcal{L})$. Though we do not show how to find lattice vectors with this length in time $2^{n/2+o(n)}$, we do show that our algorithm suffices for the most important application of such algorithms: basis reduction. In particular, we show a modified version of Gama and Nguyen's slide-reduction algorithm [Gama and Nguyen, STOC 2008], which can be combined with the algorithm above to improve the time-length tradeoff for shortest-vector algorithms in nearly all regimes, including the regimes relevant to cryptography.

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