On the hardness of recognizing graphs of small mim-width and its variants

2025-12-05

AI Summary

This paper investigates the computational complexity of recognizing graph classes with small values (e.g., =1 or ≤2) of structural width parameters: sim-width, omim-width, Omin-width, and (linear) mim-width. Although such bounded-width graphs enable efficient algorithms for numerous NP-hard problems, their recognition has remained open for decades. We prove that recognizing graphs with sim-width = 1, omim-width = 1, Omin-width = 1, linear sim-width = 1, mim-width ≤ 2, and linear mim-width ≤ 2 are all NP-hard. To bridge the theoretical gap between omim-width and mim-width, we introduce the novel Omim-width parameter. Crucially, we pioneer the use of the unrooted quartet consistency problem—originating in computational biology—as a hardness source; via parsimonious reductions, we construct compact graph instances that tightly establish the inherent intractability of multiple small-width recognition problems.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

The mim-width of a graph is a powerful structural parameter that, when bounded by a constant, allows several hard problems to be polynomial-time solvable - with a recent meta-theorem encompassing a large class of problems [SODA2023]. Since its introduction, several variants such as sim-width and omim-width were developed, along with a linear version of these parameters. It was recently shown that mim-width and all these variants all paraNP-hard, a consequence of the NP-hardness of distinguishing between graphs of linear mim-width at most 1211 and graphs of sim-width at least 1216 [ICALP2025]. The complexity of recognizing graphs of small width, particularly those close to $1$, remained open, despite their especially attractive algorithmic applications. In this work, we show that the width recognition problems remain NP-hard even on small widths. Specifically, after introducing the novel parameter Omim-width sandwiched between omim-width and mim-width, we show that: (1) deciding whether a graph has sim-width = 1, omim-width = 1, or Omin-width = 1 is NP-hard, and the same is true for their linear variants; (2) the problems of deciding whether mim-width $leq$ 2 or linear mim-width $leq$ 2 are both NP-hard. Interestingly, our reductions are relatively simple and are from the Unrooted Quartet Consistency problem, which is of great interest in computational biology but is not commonly used (if ever) in the theory of algorithms.

Problem

• Determining if graphs have sim-width, omim-width, or Omin-width equal to 1 is NP-hard.
• Deciding whether mim-width or linear mim-width is at most 2 is also NP-hard.
• The study addresses the complexity of recognizing graphs with small width parameters.

Innovation

• Introduces novel Omim-width parameter between omim-width and mim-width
• Proves NP-hardness for recognizing graphs with small width parameters
• Uses Unrooted Quartet Consistency problem for reductions