Analysis of graph coloring problem based on satisfiability and maximal independent set

Abstract

Nowadays, everywhere resource scheduling is an important task. In general, it is observed that resources are limited and users are quite more than resources; then question is how to maximize utilization of resources without conflict or with minimum conflict. The graph coloring problem is mainly used for resource scheduling. A k-colorability of graph G is an assignment of colors {1,2,…,k} to the vertices of G in such a way that neighbor vertices of graph should not receive the same colors. The minimum number of colors needed to properly color the vertices of G is called the chromatic number of G.Graph coloring problem has several important real-world applications including register allocation problem, channel assignment problem in cellular network, time tabling problem, aircraft scheduling problem, etc. Since graph coloring problem is an NP-Complete problem; therefore no exact solution could be found for large graph. There is so many heuristic algorithm used to find out approximate solution till date.This thesis presents two variations of solution approach for graph coloring problem. First is optimization based solution for graph coloring problem and second one is decision based solution for graph coloring problem. In the optimization variation of graph coloring problem, its goals to calculate the minimum possible coloursk, so that a propercolouring of graphGcould be possible. A k-colorable graph divides an array of vertices V into k dissimilarcolor classes, where each member of the class has the same color. In order to have the same color, the members of each class must be pairwise non-adjacent, which by definition makes them an independent set. In our thesis, we presented an algorithm of finding maximal independent sets from the initial graph and it gives solution for graph coloring problem.Satisfiability (SAT) is recognized as the first NP-Complete problem and one of the classic problems in computational complexity. Since, the Satisfiability problem (SAT) is interesting because it can be used as a stepping stone for solving decision problems. In our thesis, we presented Satisfiability (SAT) based solution approach for decision based graph coloring problem. In the form of a decision problem, graph k-colorability problem can be stated as follows: Is it possible to assign one of the k colors to vertices of a graph G = (V, E), such that no two adjacent nodes are assigned the same color? Ifthe answer is positive (or YES), we say that the graph is k-colorable and k is the chromatic number of graph G; otherwise it returns “unsatisfiable”. Satisfiability (SAT) is used as a starting point for proving that other problems are also NP-hard. We can reduce any NP-Complete problem to/from SAT. Therefore, in our thesis we presented a generalized reduction approach for k-colorable graph to/from 3-CNF-SATIn our thesis, we presented a polynomial 3-SAT encoding technique of k-colorable graph. This approach introduces two coloring constraint say vertex coloring constraint and edge coloring constraint for proper coloring of a graph. Since, there have been dramatic improvements in SAT solver technology over the past decade. This has lead to the development of several powerful SAT algorithms that are capable of solving many hard problems consisting of thousands of variables and millions of constraints.In thesis, we analyze an efficient SAT solver MiniSAT to investigate SAT based solution of graph k-colorability problem. Encoded 3-SAT expression will be input for SAT solver and then it gives decision based solution. In this thesis, we have analyzed the behavior of two NP-Complete problem say 3- Satisfiability and Graph 3-Colorability during reduction from each other with the help of phase transition phenomenon.Since, the channel assignment problem is very similar to the graph k-colorability problem. Reduction from graph k-colorability problem to satisfiability is an important concept to solve channel assignment in cellular network. In our thesis, we mapped a cellular network with frequency assignment and then introduced a 3-SAT encoding of channel assignment problem.