Bounds for Hardness Condensation in the Query Model
Université Paris Cité, CNRS, IRIF（巴黎西岱大学、法国国家科学研究中心、IRIF）；Indian Institute of Technology Kanpur（坎普尔印度理工学院）；Microsoft Research India（微软研究院印度分部）；Indian Institute of Technology Hyderabad（海得拉巴印度理工学院）；Indian Institute of Technology Madras（马德拉斯印度理工学院）
Chandrima Kayal, Rajat Mittal, Sai Soumya Nalli, Manaswi Paraashar, Karthikeya Polisetty, Jayalal Sarma, Nitin Saurabh
备注：
Combined version of arXiv:2602.00754 and arXiv:2602.01042. A preliminary version of this paper has been accepted for presentation at 41$^{st}$ Computational Complexity Conference (CCC 2026)}
For any Boolean function $f:\{0,1\}^n \to \{0,1\}$ with a complexity measure having value $k \ll n$, is it possible to restrict the function $f$ to $Θ(k)$ variables while keeping the complexity preserved at $Θ(k)$? This question, in the context of query complexity, was recently studied by G{ö}{ö}s, Newman, Riazanov and Sokolov (STOC 2024). They showed, among other results, that query complexity can not be condensed losslessly. They asked if complexity measures like block sensitivity or unambiguous certificate complexity can be condensed losslessly?
In this work, we show that decision tree measures like block sensitivity and certificate complexity, cannot be condensed losslessly. That is, there exists a Boolean function $f$ such that any restriction of $f$ to $O(\mathcal{M}(f))$ variables has $\mathcal{M}(\cdot)$-complexity at most $\tilde{O}(\mathcal{M}(f)^{2/3})$, where $\mathcal{M} \in \{\mathsf{bs},\mathsf{fbs},\mathsf{C},\mathsf{D}\}$. This also improves upon a result of G{ö}{ö}s, Newman, Riazanov and Sokolov (STOC 2024).
We also complement the negative results on lossless condensation with positive results about lossy condensation. In particular, we show that for every Boolean function $f$ there exists a restriction of $f$ to $O(\mathcal{M}(f))$ variables such that its $\mathcal{M}(\cdot)$-complexity is at least $Ω(\mathcal{M}(f)^{1/2})$, where $\mathcal{M} \in \{\mathsf{bs},\mathsf{fbs},\mathsf{C},\mathsf{UC}_{min},\mathsf{UC}_1,\mathsf{UC},\mathsf{D},\widetilde{\mathsf{deg}},λ\}$. We also show a slightly weaker positive result for randomized and quantum query complexity.
On the Cryptographic Hardness of Learning Single Periodic Neurons
Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）；Center for Data Science, New York University（纽约大学数据科学中心）
我们展示了一个简单的减少，这表明在存在噪声的情况下，在各向同性高斯分布上学习单个周期神经元的密码学硬度。更确切地说，我们的约简表明，在小噪声下学习这些函数的任何多项式时间算法（不一定是基于梯度的）都意味着解决最坏情况下的格问题的多项式时间量子算法，其硬度构成了基于格的密码学的基础。我们的核心硬函数族，这是很好地近似的单层神经网络，采取一般形式的一元周期函数应用于仿射投影的数据。这些函数已经出现在以前的开创性著作中，这些著作证明了它们对基于梯度（Shamir'18）和统计查询（SQ）算法（Song et al. '17）。我们表明，如果（多项式）小噪声添加到标签，学习这些功能的棘手性适用于所有多项式时间算法，超越基于梯度和SQ算法，在上述密码学假设。此外，我们证明了噪声的必要性，在硬度的结果，通过设计一个多项式时间的算法来学习某些家庭的这样的功能指数小对抗性噪声。我们提出的算法不是基于梯度或SQ算法，而是基于著名的Lenstra-Lenstra-Lovász（LLL）格基约简算法。此外，在没有噪声的情况下，该算法可以直接应用于解决CLWE检测（Bruna et al. '21）和具有$d+1$样本的最佳样本复杂度的相位恢复。在前一种情况下，这改进了在（Bruna et al. '21）。
Query Complexity of Densest Sub(hyper)graph Problems
Pacific Northwest National Laboratory（太平洋西北国家实验室）；School of Computing, University of Utah（犹他大学计算机学院）；Department of Computer Science and Engineering, Texas A&M University（德克萨斯农工大学计算机科学工程系）
Bootstrapping Classical and Quantum Query Lower Bounds for Finding Tarski Fixed Points
Center on Frontiers of Computing Studies, Peking University（前沿计算研究中心，北京大学）；School of Computer Science, Peking University（计算机学院，北京大学）；School of Electronics Engineering and Computer Science, Peking University（电子工程学院，北京大学）
