发表机构
Ben Gurion University（本-古里安大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明任意正常数概率的随机非均匀多项式规模剪枝过程若能在仿射电路输入上成功隔离满足赋值，则蕴含NP⊆P/poly，并通过区域计数组合界实现坍塌，填补了已知阈值之间的空白。
AI 中文摘要
Valiant和Vazirani以概率$\Omega(1/n)$隔离一个电路的满足赋值。Dell、Kabanets、van Melkebeek和Watanabe证明了成功率超过$2/3$蕴含$\mathrm{NP}\subseteq\mathrm{P/poly}$，并询问了介于两者之间的范围。我们证明每一个正常数已经蕴含该坍塌：如果一个随机非均匀多项式规模的剪枝过程在至多有$2^{\lfloor 2/\epsilon\rfloor}$个满足赋值的仿射电路输入上以概率$\epsilon$成功，则$\mathrm{NP}\subseteq\mathrm{P/poly}$。在至多有$L^{1/3}$个满足赋值的仿射输入上，成功率$10/\log L$就足够，其中$L$是描述长度，而在具有一个或两个满足赋值的输入上，阈值从$2/3$降至$3/5$。没有使用任何密码学假设，并且该过程可以读取整个电路。证明将一组电路编译成一个电路，其满足赋值由$\mathbb{F}_2^d$中的标签索引。每个成员被分配一个标签空间的仿射区域，如果某个成员不可满足，则满足集收缩到该成员的区域。由于区域可能重叠且具有不同维度，坍塌归结为一个组合界：没有一组标签能以超过$2/d$的比例与一个等权重的、所有维度低于$d$的仿射子空间族恰好相交于一点。这种区域计数不能低于$1/\log L$量级。由仿射哈希实现的$\Theta(1/n)$与$O(1/\log n)$之间的范围仍然开放。
英文摘要
Valiant and Vazirani isolate a satisfying assignment of a circuit with probability $Ω(1/n)$. Dell, Kabanets, van Melkebeek, and Watanabe showed that success above $2/3$ implies $\mathrm{NP}\subseteq\mathrm{P/poly}$ and asked about the range in between. We show that every positive constant already implies the collapse: if a randomized nonuniform polynomial-size pruning procedure succeeds with probability $ε$ on affine circuit inputs with at most $2^{\lfloor 2/ε\rfloor}$ satisfying assignments, then $\mathrm{NP}\subseteq\mathrm{P/poly}$. Success $10/\log L$ on affine inputs with at most $L^{1/3}$ satisfying assignments suffices, where $L$ is the description length, and on inputs with one or two satisfying assignments the threshold $2/3$ drops to $3/5$. No cryptographic assumption is used, and the procedure may read the entire circuit. The proof compiles a pool of circuits into one circuit whose satisfying assignments are indexed by tags in $\mathbb{F}_2^d$. Each member is assigned an affine region of tag space, and if one member is unsatisfiable, the satisfying set shrinks to that member's region. Because regions may overlap and have different dimensions, the collapse reduces to a combinatorial bound: no set of tags meets more than a $2/d$ fraction of an equally weighted family of affine subspaces of all dimensions below $d$ in exactly one point. This regional counting cannot go below order $1/\log L$. The range between $Θ(1/n)$, achieved by affine hashing, and $O(1/\log n)$ remains open.
