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

Max Dupré la Tour, Manuel Lafond, Ndiamé Ndiaye

TL;DR

This work investigates the hardness of recognizing graphs with very small mim-width and its variants. By constructing reductions from Unrooted Quartet Consistency, it proves NP-hardness for recognizing sim-width=1, omim-width=1, Omim-width=1 (and their linear variants), and for mim-width or linear mim-width equal to 2, while introducing Omim-width as an intermediate parameter. The results, together with ETH-based lower bounds, indicate limited prospects for efficient recognition of small-width graphs and clarify where the boundary lies for these width-based problems. The paper also discusses the tightness and limitations of the reductions and outlines open questions, including the still unresolved status of mim-width 1 recognition.

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.

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

TL;DR

This work investigates the hardness of recognizing graphs with very small mim-width and its variants. By constructing reductions from Unrooted Quartet Consistency, it proves NP-hardness for recognizing sim-width=1, omim-width=1, Omim-width=1 (and their linear variants), and for mim-width or linear mim-width equal to 2, while introducing Omim-width as an intermediate parameter. The results, together with ETH-based lower bounds, indicate limited prospects for efficient recognition of small-width graphs and clarify where the boundary lies for these width-based problems. The paper also discusses the tightness and limitations of the reductions and outlines open questions, including the still unresolved status of mim-width 1 recognition.

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 , 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 2 or linear mim-width 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.

Paper Structure

This paper contains 10 sections, 15 theorems, 7 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and Definitions
  4. Width parameters:
  5. Unrooted Quartet Consistency
  6. Sim-width, omim-width and Omim-width
  7. Remarks on the graph class of hard instances.
  8. Using a similar reduction for mim-width 1 (or linear mim-width 1)?
  9. Mim-width and linear mim-width
  10. Open problems

Key Result

Theorem 1

The problem of deciding whether a graph has sim-width, omim-width, or Omim-width equal to $1$ is NP-complete. Moreover, assuming the ETH, it cannot be solved in time $2^{o(n)}$, where $n$ is the number of vertices of the graph. The same holds for the linear variant of these three parameters.

Figures (4)

  • Figure 1: A graph illustrating the differences of the width parameters on a specific cut $(A, B)$
  • Figure 2: Subgraph of $G$ corresponding to a quartet $q = [p_i p_j \mid p_k p_{\ell}]$.
  • Figure 3: Gadget corresponding to a single quartet $q = [p_i p_j \mid p_k p_{\ell}]$, with the extra vertex $\omega$ of $H$ represented.
  • Figure 4: Expansion of $p_i$ in $T$ into the subtree corresponding to $C_i$ in $T'$.

Theorems & Definitions (34)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Theorem 3: Steel1992
  • Theorem 4
  • proof
  • Proposition 1
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 24 more