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Equitable Connected Partition and Structural Parameters Revisited: N-Fold Beats Lenstra

Authors Václav Blažej , Dušan Knop , Jan Pokorný , Šimon Schierreich

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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph algorithms
Keywords
  • Equitable Connected Partition
  • structural parameters
  • fixed-parameter tractability
  • N-fold integer programming
  • tree-width
  • shrub-depth
  • modular-width

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Abstract

In the Equitable Connected Partition (ECP for short) problem, we are given a graph G = (V,E) together with an integer p ∈ ℕ, and our goal is to find a partition of V into p parts such that each part induces a connected sub-graph of G and the size of each two parts differs by at most 1. On the one hand, the problem is known to be NP-hard in general and W[1]-hard with respect to the path-width, the feedback-vertex set, and the number of parts p combined. On the other hand, fixed-parameter algorithms are known for parameters the vertex-integrity and the max leaf number.
In this work, we systematically study ECP with respect to various structural restrictions of the underlying graph and provide a clear dichotomy of its parameterised complexity. Specifically, we show that the problem is in FPT when parameterized by the modular-width and the distance to clique. Next, we prove W[1]-hardness with respect to the distance to cluster, the 4-path vertex cover number, the distance to disjoint paths, and the feedback-edge set, and NP-hardness for constant shrub-depth graphs. Our hardness results are complemented by matching algorithmic upper-bounds: we give an XP algorithm for parameterisation by the tree-width and the distance to cluster. We also give an improved FPT algorithm for parameterisation by the vertex integrity and the first explicit FPT algorithm for the 3-path vertex cover number. The main ingredient of these algorithms is a formulation of ECP as N-fold IP, which clearly indicates that such formulations may, in certain scenarios, significantly outperform existing algorithms based on the famous algorithm of Lenstra.

Cite As Get BibTex

Václav Blažej, Dušan Knop, Jan Pokorný, and Šimon Schierreich. Equitable Connected Partition and Structural Parameters Revisited: N-Fold Beats Lenstra. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.29

Author Details

Václav Blažej
  • University of Warwick, UK
Dušan Knop
  • Czech Technical University in Prague, Czech Republic
Jan Pokorný
  • Czech Technical University in Prague, Czech Republic
Šimon Schierreich
  • Czech Technical University in Prague, Czech Republic

Funding

This work was co-funded by the European Union under the project Robotics and advanced industrial production (reg. no. CZ.02.01.01/00/22_008/0004590).
  • Blažej, Václav: Supported by the Engineering and Physical Sciences Research Council [grant number EP/V044621/1].
  • Knop, Dušan: Acknowledges the support of the Czech Science Foundation Grant No. 22-19557S.
  • Pokorný, Jan: Acknowledges the support of the Grant Agency of the CTU in Prague funded grant No. SGS23/205/OHK3/3T/18.
  • Schierreich, Šimon: Acknowledges the support of the Czech Science Foundation Grant No. 22-19557S and of the Grant Agency of the CTU in Prague funded grant No. SGS23/205/OHK3/3T/18.

Acknowledgements

We are grateful to all anonymous referees for their valuable comments.

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