（扩展）阈值维数与（半）梯索引的计算复杂性
On the Computational Complexity of (Extended) Threshold Dimension and (Semi-)Ladder Index
AI总结：
该研究证明了扩展阈值维数同时为NP-难和co-NP-难，关联最大平衡二部图变体得到两种维数的强近似难度，还证明了梯索引和半梯索引计算的近似难度，解答了开放问题并为相关算法设计提供支撑。
AI中文摘要：
我们研究假设类的阈值维数及其变体——扩展阈值维数的计算复杂性。对于后者，我们证明其同时为NP-难和co-NP-难，这（部分）解答了Dmitriev等人（SODA 2026）提出的开放问题。此外，通过将该问题与最大平衡二部图的变体关联，我们证明了两种维数的强近似难度，包括参数化情形下的难度。作为中间结果，我们还证明了梯索引和半梯索引（Fabianski等人，STACS 2019）计算的近似难度，这两种索引近期被用于设计固定参数可处理算法。
英文摘要：
We study the complexity of computing the Threshold dimension of a hypothesis class and its variant, the Extended threshold dimension. For the latter, we prove that it is both NP-hard and co-NP-hard, which (partially) answers an open question of Dmitriev et al. (SODA 2026). Furthermore, by relating the problem to a variant of Maximum Balanced Biclique, we prove strong hardness of approximation for both dimensions, including in the parameterized setting. As an intermediate result, we also prove hardness (of approximation) results for computing the ladder index and the semi-ladder index (Fabianski et al., STACS 2019), which have recently been used in the design of fixed-parameter tractable algorithms.
