Efficient Solvability of the Weighted Vertex Coloring Problem for Some Hereditary Class of Graphs with $$\boldsymbol {5}$$ -Vertex Prohibitions

2020 ◽  
Vol 14 (3) ◽  
pp. 480-489
Author(s):  
D. V. Gribanov ◽  
D. S. Malyshev ◽  
D. B. Mokeev
Keyword(s):  
Vertex Coloring ◽  
Coloring Problem ◽  
Hereditary Class ◽  
Vertex Coloring Problem ◽  
Weighted Vertex

Models and heuristic algorithms for a weighted vertex coloring problem

2008 ◽  
Vol 15 (5) ◽  
pp. 503-526 ◽  
Author(s):  
Enrico Malaguti ◽  
Michele Monaci ◽  
Paolo Toth
Keyword(s):  
Heuristic Algorithms ◽  
Vertex Coloring ◽  
Coloring Problem ◽  
Vertex Coloring Problem ◽  
Weighted Vertex

scholarly journals On new algorithmic techniques for the weighted vertex coloring problem

2020 ◽  
Vol 22 (4) ◽  
pp. 442-448
Author(s):  
Olga O. Razvenskaya
Keyword(s):  
Chromatic Number ◽  
Polynomial Time ◽  
Graph Decomposition ◽  
Vertex Coloring ◽  
Graph Reduction ◽  
Coloring Problem ◽  
Vertex Coloring Problem ◽  
Weighted Vertex ◽  
Algorithmic Techniques ◽  
Standard Techniques

The classical NP-hard weighted vertex coloring problem consists in minimizing the number of colors in colorings of vertices of a given graph so that, for each vertex, the number of its colors equals a given weight of the vertex and adjacent vertices receive distinct colors. The weighted chromatic number is the smallest number of colors in these colorings. There are several polynomial-time algorithmic techniques for designing efficient algorithms for the weighted vertex coloring problem. For example, standard techniques of this kind are the modular graph decomposition and the graph decomposition by separating cliques. This article proposes new polynomial-time methods for graph reduction in the form of removing redundant vertices and recomputing weights of the remaining vertices so that the weighted chromatic number changes in a controlled manner. We also present a method of reducing the weighted vertex coloring problem to its unweighted version and its application. This paper contributes to the algorithmic graph theory.

A DNA computer model for solving vertex coloring problem

2006 ◽  
Vol 51 (20) ◽  
pp. 2541-2549 ◽  
Author(s):  
Jin Xu ◽  
Xiaoli Qiang ◽  
Fang Gang ◽  
Kang Zhou
Keyword(s):  
Computer Model ◽  
Vertex Coloring ◽  
Coloring Problem ◽  
Vertex Coloring Problem ◽  
Dna Computer

A parallel biological computing algorithm to solve the vertex coloring problem with polynomial time complexity

2021 ◽  
pp. 1-11
Author(s):  
Zhaocai Wang ◽  
Dangwei Wang ◽  
Xiaoguang Bao ◽  
Tunhua Wu
Keyword(s):  
Time Complexity ◽  
Optimization Problems ◽  
Combinatorial Problem ◽  
Vertex Coloring ◽  
Intelligent Computing ◽  
Coloring Problem ◽  
Combinatorial Optimization Problems ◽  
Computing Algorithm ◽  
Vertex Coloring Problem ◽  
Dna Algorithm

The vertex coloring problem is a well-known combinatorial problem that requires each vertex to be assigned a corresponding color so that the colors on adjacent vertices are different, and the total number of colors used is minimized. It is a famous NP-hard problem in graph theory. As of now, there is no effective algorithm to solve it. As a kind of intelligent computing algorithm, DNA computing has the advantages of high parallelism and high storage density, so it is widely used in solving classical combinatorial optimization problems. In this paper, we propose a new DNA algorithm that uses DNA molecular operations to solve the vertex coloring problem. For a simple n-vertex graph and k different kinds of color, we appropriately use DNA strands to indicate edges and vertices. Through basic biochemical reaction operations, the solution to the problem is obtained in the O (kn2) time complexity. Our proposed DNA algorithm is a new attempt and application for solving Nondeterministic Polynomial (NP) problem, and it provides clear evidence for the ability of DNA calculations to perform such difficult computational problems in the future.

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