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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21240%2F24%3A00375730" target="_blank" >RIV/68407700:21240/24:00375730 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.4230/LIPIcs.MFCS.2024.29" target="_blank" >https://doi.org/10.4230/LIPIcs.MFCS.2024.29</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.4230/LIPIcs.MFCS.2024.29" target="_blank" >10.4230/LIPIcs.MFCS.2024.29</a>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

In the Equitable Connected Partition (ECP for short) problem, we are given a graph $G=(V,E)$ together with an integer $pinmathbb{N}$, 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.

Název v anglickém jazyce

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

Popis výsledku anglicky

In the Equitable Connected Partition (ECP for short) problem, we are given a graph $G=(V,E)$ together with an integer $pinmathbb{N}$, 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.

Druh

D - Stať ve sborníku

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

Výsledek vznikl pri realizaci vícero projektů. Více informací v záložce Projekty.

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2024

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

Proceedings of the 49th International Symposium on Mathematical Foundations of Computer Science

ISBN

978-3-95977-335-5

ISSN

1868-8969

e-ISSN

—

Počet stran výsledku

16

Strana od-do

"29:1"-"29:16"

Název nakladatele

Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik

Místo vydání

—

Místo konání akce

Bratislava

Datum konání akce

26. 8. 2024

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

—