We study the problem of locating violating principal minors in structured matrix families that lie near the boundary of P-matrices and admit sparse violations under perturbation. Viewing violation search as an information acquisition problem, we show that, despite strong underlying structure, the location of a violation may be globally encoded and not accessible through local queries under a restricted interaction model. This leads to an information-theoretic bottleneck: each query reveals only vanishing information about the violating subset, so that polynomially many queries accumulate insufficient information to identify it. Using mutual information and Fano's inequality, we show that any algorithm restricted to polynomially many queries cannot recover the violating subset with constant success probability. Our analysis highlights a distinction between structure and accessibility: even highly structured problems can be computationally intractable when the information required to locate a solution is not accessible through the available queries.

翻译：我们研究位于P-矩阵边界附近、且在扰动下允许稀疏违规的结构化矩阵族中违规主次子式的定位问题。将违规搜索视为信息获取问题，我们证明：尽管存在强底层结构，但在受限交互模型下，违规位置可能被全局编码而无法通过局部查询访问。这导致信息论瓶颈：每次查询仅揭示关于违规子集的零散信息，因此多项式次查询积累的信息不足以识别该子集。利用互信息和法诺不等式，我们证明任何受限于多项式次查询的算法都无法以恒定成功概率恢复违规子集。我们的分析凸显了结构与可达性之间的区别：当定位解所需的信息无法通过可用查询访问时，即使高度结构化的问题也可能计算上不可处理。