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 is globally encoded and not accessible through local queries. 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 fundamental 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-矩阵边界附近且扰动下存在稀疏违例的结构化矩阵族中违规主子的定位问题。将违例搜索视为信息获取问题，我们证明尽管存在强结构特性，但违例位置的全局编码特性使其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。利用互信息与法诺不等式，我们证明任何受限于多项式次查询的算法无法以恒定成功概率恢复违规子集。分析揭示了结构与可达性之间的根本区别：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在计算上不可处理。