Parameterized Hardness of Mixed 2-Sided Orthant Depth

2026-09-30

AI Summary

This study addresses the mixed two-sided orthogonal range maximum depth problem in high-dimensional spaces, which seeks a point in $\mathbb{R}^d$ covered by the maximum number of two-sided constrained regions. Methodologically, leveraging parameterized complexity theory, we establish through a polynomial-time reduction from the MultiColoredClique problem that the corresponding decision problem is W[1]-hard when parameterized by the dimension $d$. This contribution rigorously defines a parameterized computational hardness lower bound for the problem with respect to dimensionality, revealing its intrinsic intractability in high-dimensional settings. Furthermore, it demonstrates that, under standard complexity assumptions, no efficient fixed-parameter tractable (FPT) algorithm exists for this problem.

Abstract

We consider the maximum-depth problem for mixed 2-sided orthants in R^d: each region imposes one lower bound and one upper bound on distinct coordinates, and the task is to find a point contained in as many regions as possible. We show that the corresponding decision problem is W[1]-hard when parameterized by the dimension. Our reduction from MultiColoredClique uses two coordinates per color class and only polynomially many orthants.

Problem

• Maximum-depth problem
• Mixed 2-sided orthants
• Parameterized hardness
• W[1]-hard
• Computational geometry

Innovation

• Parameterized Hardness
• W[1]-hard
• Mixed 2-Sided Orthant Depth
• MultiColoredClique Reduction
• Computational Geometry