Ap Problem Formulation For The Traveling Salesman Problem

A. Travelling Salesman Problem

The traveling salesman problem is standout among the most studied well-known problems. Given a set of cities, the problem is finding the shortest route that visits all cities and returns back to the origin point. One possible way to solve this would be to compute all possible routes and choose the best. In the past decade, several ways have been dedicated to solve computationally hard optimization problems like TSP. TSP is an NP- Complete which means there is no exact algorithm to solve it ideally in polynomial time [1]. They’re basically attempt to generate an optimal or near-optimal solutions in terms of running time cost and space requirements.
Fortunately, many successful approximation methods have been

TSP Problem Formulation

In Traveling salesman problem, the distance between each city is identical in each opposite direction, and that will create undirected graph.
The best direct solution for the TSP would be to find the shortest route by complete edge -weighted graph G with N nodes or vertices representing the cities and N(N-1) edges that represented the paths. Each path has a rate that represents the distance to get from one vertex to another one [3].
Formally, the TSP can be stated as:
The distances between N cities are stored in a ‘cost matrix’ with (dimension nxn), where n represent the total number of nodes of the undirected graph G.
By given C = (cij), where (i, j) denoted the pairs of the nodes.
(cij) denotes the cost of moving from city i to city j, where
(i, j = 1, ..., n), and the diagonal elements (cij )=0
A tour can be represented by a cyclic permutation of (i1, i2, i3, ..., in) of the integers from 1 through n that minimizes the size ci1 i2 + ci2 i3 + ... + cin i1. [4]
The challenge in the problem is that the traveling salesman wants to establish the minimum total length of Hamiltonian cycle in both time and cost.

Fig.1: Travelling Salesman Problem

C. The Basic Greedy

The basic idea of the algorithm described as follows:
By given a set of (E, F), where E is denoted as a finite set and F is the group of subsets of E [5].
C is the weight function that responsible to allocate a real weight C(e) to each component of E. The sum of weights of each component of the S can be defined as C(S), where S ∈ F. A maximal set with the minimal weight need to be considered where (B ∈ F), therefore, we will consider only optimization problems called (E, F, W) Let I(S) = {x: S ∪ {x} ∈ F} – S, where S ∈ F and I(S) involves those components from E – S and added them to S to have an independent set of size |S| + 1.
Considering I(S) ≠ ∅ for each independent set S, the greedy algorithm will try to make a minimum weight as follows:
Step1: Starting from an empty set M.
Step2: At each step will take the current set M and add a minimum weight element to it, where the element e ∈ I(X)
Step 3: The algorithm will stop when a minimum weight base B is completely constructed.
Base B will build to provide a suitable choice between components of the same weight are made.

Fig.2: The concept of Greedy algorithm

II. RELATED