Research of NP-Complete Problems in the Class of Prefractal Graphs

Rasul Kochkarov

doi:10.3390/math9212764

Abstract

NP-complete problems in graphs, such as enumeration and the selection of subgraphs with given characteristics, become especially relevant for large graphs and networks. Herein, particular statements with constraints are proposed to solve such problems, and subclasses of graphs are distinguished. We propose a class of prefractal graphs and review particular statements of NP-complete problems. As an example, algorithms for searching for spanning trees and packing bipartite graphs are proposed. The developed algorithms are polynomial and based on well-known algorithms and are used in the form of procedures. We propose to use the class of prefractal graphs as a tool for studying NP-complete problems and identifying conditions for their solvability. Using prefractal graphs for the modeling of large graphs and networks, it is possible to obtain approximate solutions, and some exact solutions, for problems on natural objects—social networks, transport networks, etc.

Highlights

The beginning of the study of intractable problems is associated with the possibility of eliminating enumeration to find the optimal solution and create a polynomial algorithm
A theorem is proposed for the isomorphism of two prefractal graphs, provided that the adjacency of the old edges is preserved
We study only a small part of the known NP-complete problems

Summary

Introduction

The beginning of the study of intractable problems is associated with the possibility of eliminating enumeration to find the optimal solution and create a polynomial algorithm. In the case of an exponential number of variants of solutions, the exhaustive algorithm does not allow one to find the optimal solution in an acceptable time and becomes intractable or even unsolvable. The efficiency of finding a solution and the complexity of the algorithm are estimated based on the solution time, limited by a function of the size of the problem. In the NP class, typical (NP-complete) problems that set the “standard” of complexity are distinguished. Any problem from NP reduces polynomially to an NP-complete problem [1]

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