Learning-Based Approaches for Graph Problems: A Survey

Over the years, many graph problems specifically those in NP-complete are studied by a wide range of researchers. Some famous examples include graph colouring, travelling salesman problem and subgraph isomorphism. Most of these problems are typically addressed by exact algorithms, approximate algorithms and heuristics. There are however some drawback for each of these methods. Recent studies have employed learning-based frameworks such as machine learning techniques in solving these problems, given that they are useful in discovering new patterns in structured data that can be represented using graphs. This research direction has successfully attracted a considerable amount of attention. In this survey, we provide a systematic review mainly on classic graph problems in which learning-based approaches have been proposed in addressing the problems. We discuss the overview of each framework, and provide analyses based on the design and performance of the framework. Some potential research questions are also suggested. Ultimately, this survey gives a clearer insight and can be used as a stepping stone to the research community in studying problems in this field.Comment: v1: 41 pages; v2: 40 page

On the diophantine equation x² + 4.7ᵇ = y²ʳ

This paper investigates and determines the solutions for the Diophantine equation x²+ 4.7ᵇ= y²ͬ, where x, y, bare all positive intergers and r> 1. By substituting the values of rand b respectively, generators of x and yͬ can be determined and classified into different categories. Then, by using geometric progression method, a general formula for each category can be obtained. The necessary conditions to obtain the integral solutions of x and y are also investigated

Discovering factors of graph polynomials

One of the most common approaches in studying any polynomial is by looking at its factors. Over the years, different graph polynomials have been defined for both undirected and directed graphs, including the Tutte polynomial, chromatic polynomial, greedoid polynomial and cover polynomial. We consider two graph polynomials, one for undirected graphs and one for directed graphs. We first give an overview of these two polynomials. We then discuss the factors of these polynomials as well as the information that are encapsulated by these factors

Integral Solution to the Equation x2+2a.7b=yn

Diophantine equation is an equation in which solutions to it are from some predetermined classes and it is one of the oldest branches of number theory. There are many types of diophantine equations, for instance linear diophantine equation, exponential diophantine equation and others. In this research, we will investigate and find the integral solutions to the diophantine equation x² +2a.7b=yⁿ where a and b are positive integers and n is even. By fixing n = 2r , we determine the generators of and x and yr for 1 ≤ a ≤ 6 with any values of b. Then, we investigate the necessary conditions to obtain integral solutions of x and y under each value of af there is any. The approach is by looking at the possible combinations for the product 2a⋅7b and solving the equations simultaneously. Then, from the results obtained, we substitute the values of a followed by b to get integer values of and a and yr under each category. After that, the equations are grouped according to the pattern that emerged and a geometric progression formula is applied to create the general formulae for the generators of solutions to the equation examined. Besides that, we have to identify the range of i, the number of non-negative integral solutions associated with each b for different values of a. When b is even, we find some special cases of determining the generators of solutions for and x and yr with a certain condition. From our investigation, we find that there is no integral solution of and x and yr the diophantine equation x² +2a.7b=yⁿ, when n is even and a = 1. It is found that the number of generators to determine the integral solutions to the equation depend on the values of a. Values of y are determined by taking the r-th root of yr for certain values of r

The winning percentage in congkak using a randomised strategy

Congkak is a traditional counting game played in Southeast Asia including Malaysia, Singapore, Brunei and Indonesia. To start a game, a board that has 16 holes together with 98 marbles are required. Each player controls a set of seven holes and own a store. The winner of the game is the player who captured more marbles into the store at the end of the game. Note that the firstmove advantage exists in chess; we investigate if the first-move advantage holds in congkak also. We model the route for each player in congkak using a directed graph and adopt these graph representations in our programs, to compute the winning percentage of each player. We focus on games between novices, hence a randomised strategy is used in our algorithm. We present the first experimental results for 100,000 games between novices in congkak. We also suggest some questions for future research in this area

Gaussian Integer Solutions of the Diophantine Equation x^4+y^4=z^3 for x≠ y

The investigation of determining solutions for the Diophantine equation  over the Gaussian integer ring for the specific case of  is discussed. The discussion includes various preliminary results later used to build the resolvent theory of the Diophantine equation studied. Our findings show the existence of infinitely many solutions. Since the analytical method used here is based on simple algebraic properties, it can be easily generalized to study the behavior and the conditions for the existence of solutions to other Diophantine equations, allowing a deeper understanding, even when no general solution is known