Iterated Type Partitions

01/22/2020
by   Gennaro Cordasco, et al.
0

This paper deals with the complexity of some natural graph problems when parametrized by measures that are restrictions of clique-width, such as modular-width and neighborhood diversity. The main contribution of this paper is to introduce a novel parameter, called iterated type partition, that can be computed in polynomial time and nicely places between modular-width and neighborhood diversity. We prove that the Equitable Coloring problem is W[1]-hard when parametrized by the iterated type partition. This result extends to modular-width, answering an open question about the possibility to have FPT algorithms for Equitable Coloring when parametrized by modular-width. Moreover, we show that the Equitable Coloring problem is instead FTP when parameterized by neighborhood diversity. Furthermore, we present simple and fast FPT algorithms parameterized by iterated type partition that provide optimal solutions for several graph problems; in particular, this paper presents algorithms for the Dominating Set, the Vertex Coloring and the Vertex Cover problems. While the above problems are already known to be FPT with respect to modular-width, the novel algorithms are both simpler and more efficient: For the Dominating set and Vertex Cover problems, our algorithms output an optimal set in time O(2^t+poly(n)), while for the Vertex Coloring problem, our algorithm outputs an optimal set in time O(t^2.5t+o(t)log n+poly(n)), where n and t are the size and the iterated type partition of the input graph, respectively.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
01/12/2022

Polynomial Turing Compressions for Some Graph Problems Parameterized by Modular-Width

In this paper we investigate the parameterized complexity for NP-hard gr...
research
07/05/2023

Parameterized Complexity of Domination Problems Using Restricted Modular Partitions

For a graph class 𝒢, we define the 𝒢-modular cardinality of a graph G as...
research
07/08/2021

An Efficient Reduction of a Gammoid to a Partition Matroid

Our main contribution is a polynomial-time algorithm to reduce a k-color...
research
03/09/2020

b-Coloring Parameterized by Clique-Width

We provide a polynomial-time algorithm for b-Coloring on graphs of const...
research
06/24/2020

Acyclic coloring of special digraphs

An acyclic r-coloring of a directed graph G=(V,E) is a partition of the ...
research
11/06/2017

Applying Convex Integer Programming: Sum Multicoloring and Bounded Neighborhood Diversity

In the past 30 years, results regarding special classes of integer linea...
research
04/10/2023

Geometry of Rounding: Near Optimal Bounds and a New Neighborhood Sperner's Lemma

A partition 𝒫 of ℝ^d is called a (k,ε)-secluded partition if, for every ...

Please sign up or login with your details

Forgot password? Click here to reset