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：我们研究在结构化矩阵族中定位违规主子的子问题，这些矩阵族位于P-矩阵边界附近，且在扰动下允许稀疏违规。将违规搜索视为信息获取问题，我们证明：尽管存在强潜在结构，但在受限交互模型下，违规位置可能全局编码且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渐近消失信息，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Fano不等式，我们证明任何限制多项式次查询的算法无法以恒定成功概率恢复违规子集。分析突显了结构与可达性之间的区别：当定位解所需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棘手的。