On the Computational Complexity of (Extended) Threshold Dimension and (Semi-)Ladder Index

2026-07-30

AI Summary

This study investigates the computational complexity of the extended threshold dimension and (semi-)stepwise index. By reducing the problem to a variant of the Maximum Balanced Biclique problem, it establishes for the first time that computing the extended threshold dimension is both NP-hard and co-NP-hard. Strong hardness results are further derived within the frameworks of approximation and parameterized complexity. The work resolves an open question posed by Dmitriev et al., provides theoretical limits for recent fixed-parameter tractable (FPT) algorithms based on the stepwise index, and reveals the inherent computational intractability of related graph structural parameters.

Abstract

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.

Problem

• Threshold dimension
• Extended threshold dimension
• ladder index
• semi-ladder index
• computational complexity

Innovation

• Threshold dimension
• Computational complexity
• Hardness of approximation
• Parameterized complexity
• Ladder index