Core Concepts

A counterexample has been found that invalidates a crucial lemma in the ICALP 2008 paper on a linear-time modular decomposition algorithm, rendering the algorithm incorrect.

Abstract

The authors provide a counterexample graph G that produces an incorrect output when processed by the ICALP 2008 modular decomposition algorithm. They trace the flaw to Lemma 4 in the original paper, which is shown to be false on the example graph.
The algorithm starts by choosing a vertex x as the first pivot and recursively processes the neighbors G(x) and non-neighbors G(x) of x. During the refinement step, the algorithm uses Lemma 4 to identify the strong modules not containing x. However, the counterexample demonstrates that the lemma does not hold, as the strong module T1 not containing x has marked children, contradicting the lemma.
This fundamental issue with the algorithm's correctness proof means that the implementation based on this algorithm will also produce incorrect results, as observed by the authors. The authors contacted the original authors, who acknowledged the problem but did not mention it in their subsequent revision of the paper.
The main purpose of this note is to publicize the problem with the ICALP 2008 algorithm and its associated implementation to prevent further waste of time on attempts to implement the incorrect algorithm.

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by William Athe... at **arxiv.org** 04-23-2024

Deeper Inquiries

To develop a correct linear-time modular decomposition algorithm, alternative approaches or techniques could be explored:
LexBFS Algorithm: As seen in the revised work by Corneil, Habib, Paul, and Tedder, utilizing LexBFS can offer a different path to achieving linear-time modular decomposition. This approach could potentially address the issues encountered in the ICALP 2008 algorithm.
Dynamic Programming: Implementing dynamic programming techniques tailored to modular decomposition could lead to a more robust and accurate algorithm. By breaking down the problem into smaller subproblems and storing intermediate results, the algorithm's correctness can be ensured.
Graph Theory Concepts: Leveraging advanced graph theory concepts such as treewidth, tree decompositions, or chordal graphs may provide insights into developing a more reliable modular decomposition algorithm. These concepts can offer structural properties that aid in efficient decomposition.

To address the issues raised in this note and provide a comprehensive correction or revision of their work, the authors of the ICALP 2008 paper can take the following steps:
Acknowledgment and Correction: The authors should publicly acknowledge the identified problem and its implications. They should issue a formal correction or revision to their original paper, clearly outlining the errors and providing corrected algorithms or proofs.
Collaboration and Peer Review: Engage with the research community, especially experts in graph theory and algorithm design, to review the revised work thoroughly. Collaborating with peers can help in identifying any remaining issues and ensuring the accuracy of the revised algorithm.
Publication of Errata: Publish an errata or supplementary document detailing the corrections made to the ICALP 2008 paper. This document should explain the corrected algorithm, address the faulty lemma, and provide insights into the revised approach taken to achieve linear-time modular decomposition.

The counterexample presented in this note has broader implications for the field of graph theory and algorithm design:
Learning Opportunity: The community can learn from this experience by understanding the importance of rigorous testing, peer review, and validation in algorithmic research. It highlights the need for thorough verification of algorithms, especially in complex domains like graph theory.
Algorithmic Integrity: Emphasizes the significance of algorithmic integrity and the impact of incorrect assumptions or lemmas on the overall algorithm. Researchers should prioritize robustness and correctness in algorithm design to avoid propagating flawed solutions.
Advancing Research: This experience can drive further research in modular decomposition algorithms, encouraging the exploration of new methodologies, validation techniques, and algorithmic paradigms. It underscores the iterative nature of algorithm development and the continuous quest for improved solutions in graph theory.

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