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Decline and Fall of the ICALP 2008 Modular Decomposition algorithm

William Atherton, Dmitrii V. Pasechnik

TL;DR

The work examines the correctness of the ICALP 2008 linear-time modular decomposition method and presents a concrete counterexample graph $G$ that invalidates a crucial lemma. The counterexample shows that the refinement and promotion steps can mark children of a strong module not containing $x$, contradicting the lemma’s claim and leading to a missing prime node. The flaw is traced to Lemma 4 (Lemma 3.1 in the preprint), including a typographical issue ('not') that hides the true falsity; the example demonstrates the incorrect behavior on $G$ and on its complement $ackslashoverline{G}$. As a result, the authors note that Corneil 2024 adopts a LexBFS-based revision, underscoring the need for alternative strategies in linear-time modular decomposition.

Abstract

We provide a counterexample to a crucial lemma in the ICALP 2008 paper "Simpler Linear-Time Modular Decomposition Via Recursive Factorizing Permutations", invalidating the algorithm described there.

Decline and Fall of the ICALP 2008 Modular Decomposition algorithm

TL;DR

The work examines the correctness of the ICALP 2008 linear-time modular decomposition method and presents a concrete counterexample graph that invalidates a crucial lemma. The counterexample shows that the refinement and promotion steps can mark children of a strong module not containing , contradicting the lemma’s claim and leading to a missing prime node. The flaw is traced to Lemma 4 (Lemma 3.1 in the preprint), including a typographical issue ('not') that hides the true falsity; the example demonstrates the incorrect behavior on and on its complement . As a result, the authors note that Corneil 2024 adopts a LexBFS-based revision, underscoring the need for alternative strategies in linear-time modular decomposition.

Abstract

We provide a counterexample to a crucial lemma in the ICALP 2008 paper "Simpler Linear-Time Modular Decomposition Via Recursive Factorizing Permutations", invalidating the algorithm described there.
Paper Structure (4 sections, 1 theorem, 1 equation, 1 figure, 2 algorithms)

This paper contains 4 sections, 1 theorem, 1 equation, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. The example
  3. Faulty lemma
  4. Conclusion

Key Result

Lemma 1

The nodes in the ordered list of trees resulting from refinement that do not have marked children correspond exactly to the strong modules not containing $x$.

Figures (1)

  • Figure 1: The modular decomposition of $G$, with "series", resp. "parallel" (abbreviated "//"), resp. "prime", nodes of the decomposition tree are show by , resp. , resp. boxes.

Theorems & Definitions (2)

  • Lemma : Lemma 4 in 10.1007/978-3-540-70575-8_52
  • Definition 1: 10.1007/978-3-540-70575-8_52