单因子稀疏投资组合选择的结构复杂性：精确算法、参数化困难性与受限电路下界
Structural Complexity of One-Factor Sparse Portfolio Selection: Exact Algorithms, Parameterized Hardness, and Restricted Circuit Lower Bounds
- Business and Technology University (BTU)（商业与技术大学）
- Eurasian Logistics Research Center（欧亚物流中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究单因子模型下等权重最小方差投资组合的稀疏选择，证明一般情形NP完全且参数化困难，给出伪多项式算法，并证明在受限电路下不属于AC^0。
AI中文摘要：
我们在因子形式给出的单因子协方差模型下，研究精确基数、等权重的方差最小化投资组合选择问题。在非负同方差情形下，选择K个最小的因子载荷是最优的。允许严格为正的资产特异性异方差，即使载荷为正整数且协方差矩阵严格正定，该决策问题也是NP完全的；当残差协方差为单位阵时，仅需一个负载荷也足以导致NP完全性。我们针对单因子和固定因子维度给出了精确的伪多项式动态规划算法，并证明了以K为参数的W[1]困难性，包括正数据族。因此，针对Monge（2017）的等权重单因子方差输入公式，若存在一般的精确多项式时间算法，则意味着P=NP。对于归一化的二进制因子编码，我们构造了一个来自模k-SUM的深度为零的投影，该投影保持精确基数和正定性。该投影在所述宽度和量词条件下传递了Lin（2026）的固定k电路下界，并独立地给出了一个基于奇偶性的证明，表明即使残差协方差为单位阵且整数系数多项式有界，投资组合语言也不属于非一致AC^0。这些是受限电路结果：我们不声称无限制的P/poly下界或P与NP的分离。
英文摘要：
We study exact-cardinality, equally weighted minimum-variance portfolio selection under a one-factor covariance model supplied in factor form. In the nonnegative homoskedastic regime, selecting the K smallest loadings is optimal. Allowing strictly positive asset-specific idiosyncratic variances makes the decision problem NP-complete even with positive integer loadings and a strictly positive-definite covariance matrix; with identity residual covariance, exactly one negative loading also suffices. We give exact pseudo-polynomial dynamic programs for one factor and fixed factor dimension and prove W[1]-hardness parameterized by K, including the positive-data family. Consequently, a general exact polynomial-time algorithm for Monge's (2017) equally weighted single-factor variance-input formulation would imply P=NP. For a normalized binary factor encoding, we construct a depth-zero projection from modular k-SUM that preserves exact cardinality and positive definiteness. The projection transfers Lin's (2026) fixed-k circuit lower bound under its stated width and quantifier conditions and, independently, yields a parity-based proof that the portfolio language is not in nonuniform AC^0 even with identity residual covariance and polynomially bounded integer coefficients. These are restricted-circuit results: no unrestricted P/poly lower bound and no separation of P from NP is claimed.
