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# Efficient and Adaptive Parameterized Algorithms on Modular Decompositions

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LIPIcs.ESA.2018.55.pdf
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## Cite As

Stefan Kratsch and Florian Nelles. Efficient and Adaptive Parameterized Algorithms on Modular Decompositions. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 55:1-55:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ESA.2018.55

## Abstract

We study the influence of a graph parameter called modular-width on the time complexity for optimally solving well-known polynomial problems such as Maximum Matching, Triangle Counting, and Maximum s-t Vertex-Capacitated Flow. The modular-width of a graph depends on its (unique) modular decomposition tree, and can be computed in linear time O(n+m) for graphs with n vertices and m edges. Modular decompositions are an important tool for graph algorithms, e.g., for linear-time recognition of certain graph classes. Throughout, we obtain efficient parameterized algorithms of running times O(f(mw)n+m), O(n+f(mw)m) , or O(f(mw)+n+m) for low polynomial functions f and graphs of modular-width mw. Our algorithm for Maximum Matching, running in time O(mw^2 log mw n+m), is both faster and simpler than the recent O(mw^4n+m) time algorithm of Coudert et al. (SODA 2018). For several other problems, e.g., Triangle Counting and Maximum b-Matching, we give adaptive algorithms, meaning that their running times match the best unparameterized algorithms for worst-case modular-width of mw=Theta(n) and they outperform them already for mw=o(n), until reaching linear time for mw=O(1).

## Subject Classification

##### ACM Subject Classification
• Mathematics of computing → Graph algorithms
##### Keywords
• efficient parameterized algorithms
• modular-width

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