Abstract
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.
Highlights
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
We 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
Summary
В работе рассматриваются только обыкновенные графы, т. е. непомеченные, неориентированные графы без петель и кратных ребер. Вершинной раскраской графа G = (V, E) называется произвольное отображение c : V −→ N такое, что c(v1) ̸= c(v2) для любых смежных вершин v1, v2 ∈ V. Хроматическое число графа опредиляется как наименьшее количество множеств попарно несмежных вершин (независимых множеств), на которые можно разбить множество его вершин. Задача о вершинной раскраске (задача ВР) для заданного графа состоит в вычислении его хроматического числа. Для взвешенного графа (G, w) задача о взвешенной вершинной раскраске (далее, кратко задача ВВР) состоит в нахождении минимального числа k, обозначаемого через χw(G), такого, что существует функция c : V −→ 2{1,2,...,k}, где |c(v)| = w(v) для любой v ∈ V и c(v1) ∩ c(v2) = ∅ для любого ребра v1v2 графа G. О новых алгоритмических приемах для задачи о взвешенной вершинной раскраске.
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