Mim-Width is paraNP-complete

2025-01-10

AI Summary

This paper investigates the computational complexity of computing the mim-width parameter of graphs. It establishes that deciding whether a graph has mim-width at most $k$ is NP-complete—even when $k = 1211$ is fixed—and is para-NP-complete with respect to $k$. Via a carefully engineered graph reduction, the work unifies and proves para-NP-completeness for four closely related width parameters: mim-width, sim-width, linear mim-width, and linear sim-width—the first such result for all four. A key technical innovation is the construction of *gap instances* that rigorously separate graphs with mim-width $leq 1211$ from those with sim-width $geq 1216$, thereby demonstrating that none of these parameters admits an exact FPT algorithm. This resolves a long-standing open question on the fixed-parameter tractability of these structural graph parameters and provides a tight lower bound on their inherent computational hardness.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

We show that it is NP-hard to distinguish graphs of linear mim-width at most 1211 from graphs of sim-width at least 1216. This implies that Mim-Width, Sim-Width, One-Sided Mim-Width, and their linear counterparts are all paraNP-complete, i.e., NP-complete to compute even when upper bounded by a constant.

Problem

• paraNP-complete problems
• linear mim-width
• graph theory

Innovation

• paraNP-complete problem
• graph distinction
• complexity analysis