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

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.

On new algorithmic techniques for the weighted vertex coloring problem

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.

The power of locality: Exploring the limits of randomness in distributed computing

Many modern systems are built on top of large-scale networks like the Internet. This article provides an overview of a dissertation [29] that addresses the complexity of classic graph problems like the vertex coloring problem in such networks. It has been known for a long time that randomization helps significantly in solving many of these problems, whereas the best known deterministic algorithms have been exponentially slower. In the first part of the dissertation we use a complexity theoretic approach to show that several problems are complete in the following sense: An efficient deterministic algorithm for any complete problem would imply an efficient algorithm for all problems that can be solved efficiently with a randomized algorithm. Among the complete problems is a rudimentary looking graph coloring problem that can be solved by a randomized algorithm without any communication. In further parts of the dissertation we develop efficient distributed algorithms for several problems where the most important problems are distributed versions of integer linear programs, the vertex coloring problem and the edge coloring problem. We also prove a lower bound on the runtime of any deterministic algorithm that solves the vertex coloring problem in a weak variant of the standard model of the area.

Lower bounds on approximating some variations of vertex coloring problem over restricted graph classes

A [Formula: see text] vertex coloring of a graph [Formula: see text] partitions the vertex set into [Formula: see text] color classes (or independent sets). In minimum vertex coloring problem, the aim is to minimize the number of colors used in a given graph. Here, we consider three variations of vertex coloring problem in which (i) each vertex in [Formula: see text] dominates a color class, (ii) each color class is dominated by a vertex and (iii) each vertex is dominating a color class and each color class is dominated by a vertex. These minimization problems are known as Min-Dominator-Coloring, Min-CD-Coloring and Min-Domination-Coloring, respectively. In this paper, we present approximation hardness results for these problems for some restricted class of graphs.