Problemas de particionamento de grafos em árvores monocromáticas

In this work, we study the Partitioning Graphs into Monochromatic Trees (PGMT) problem. In this problem, an edge-coloured graph G with n vertices is given, and the goal is to find the smallest number of vertex disjoint monochromatic trees that cover all the vertices of G. First, we study the computa...

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Bibliographic Details
Author: Diego Rangel Piranga Costa
Format: master thesis
Status:Published version
Publication Date:2021
Country:Brasil
Institution:Universidade Federal de Minas Gerais (UFMG)
Repository:Repositório Institucional da UFMG
Language:Portuguese
OAI Identifier:oai:repositorio.ufmg.br:1843/82526
Online Access:http://hdl.handle.net/1843/82526
Access Level:Open access
Keyword:Algoritmos
Árvores monocromáticas
Complexidade parametrizada
Grafos
Computação – Teses
Algoritmos de computador – Teses
Complexidade computacional – Teses
Árvores (Teoria dos grafos) - Teses
Description
Summary:In this work, we study the Partitioning Graphs into Monochromatic Trees (PGMT) problem. In this problem, an edge-coloured graph G with n vertices is given, and the goal is to find the smallest number of vertex disjoint monochromatic trees that cover all the vertices of G. First, we study the computational complexity of this problem, in which we show that the PGMT is NP-complete when we consider some parameters such as: color frequency, maximum degree and number of colors; or when we restrict to the class of complete bipartite graphs where the number of trees is limited. We also show a lower bound for executing exact algorithms using the Exponential Time Hypothesis (ETH). More precisely, we show that there is δ > 0 such that PGMT cannot be resolved in time O(2δn), unless a ETH is false. As positive results, we present an algorithm of complexity O(n 2 ) when G is a tree and we also present an algorithm parameterized by the number of colors r and by the treewidth t of the input graph that runs in time O(n O(1)(r · t) 2t+1).

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