Question

Show that the reduction of the Traveling Salesperson (Undirected) Decision problem to the Traveling Salesperson Decision problem can be done in polynomial time.

Short answer

The reduction from the Traveling Salesperson (Undirected) Decision problem to the Traveling Salesperson Decision problem can be done in polynomial time. The key step is to replace each undirected edge with two directed edges in the opposite direction with the same weight, and this process has a time complexity of \(O(E)\), which is polynomial.

Step-by-step solution

Understand the problem

In the Traveling Salesperson Problem (TSP), we are given a weighted graph and need to find a minimum weight Hamiltonian cycle. In the decision version of TSP, it is asked whether there exists a path that visits every vertex exactly once with a total weight no more than some constant \(K\). In the standard TSP, the graph is directed, while in the undirected TSP, the graph is undirected.

Conversion of undirected graph to directed graph

Given an undirected graph of the TSUD problem, we can convert it into a directed graph for the TSD problem by replacing each undirected edge \((u, v)\) with two directed edges \((u, v)\) and \((v, u)\) each with the same weight as the undirected edge. This results in a directed graph that preserves all the properties of the original undirected graph necessary for solving the TSP.

Demonstrate polynomial time complexity

The reduction from the TSUD to the TSD, involves going through each edge once and thus has a time complexity of \(O(E)\), where \(E\) is the number of edges in the graph. As the number of edges in a graph is polynomial in terms of the number of nodes, therefore, this reduction can be done in polynomial time.

Key concepts

Graph Theory

Imagine a spider web, where the junctions are places you might want to visit, and the strands connecting them are the paths you can take. This web, in the realm of mathematics, is what we call a graph.

In graph theory, which is a fundamental part of combinatorics and computer science, we study these webs or 'graphs'. A graph consists of 'vertices' (also called nodes) which represent the junctions, and 'edges' that represent the strands or paths connecting these vertices. When you travel from one vertex to another using an edge, it's akin to taking a particular route from one destination to another in real life.

There are two types of graphs: directed and undirected. In an undirected graph, edges have no direction, much like a two-way street where you can travel in either direction. In contrast, in a directed graph, each edge is a one-way street, pointing from one vertex to another. In the context of the Traveling Salesperson Problem, understanding the distinction between these two types of graphs is crucial, as the way we approach finding a solution depends on the graph's nature.

Hamiltonian Cycle

Now, let's dive deeper into our spider web and find a particular kind of 'loop' that a very ambitious spider might trace. This loop is what mathematicians refer to as a Hamiltonian cycle.

A Hamiltonian cycle in a graph is a special path that visits each vertex exactly once and returns to the starting vertex, creating a closed loop. Think of it as a treasure hunt in which you have to visit a series of locations without visiting the same place twice and then return to where you started.

Identifying a Hamiltonian cycle is a key component in solving the Traveling Salesperson Problem (TSP), as it represents an ideal route that visits every city (vertex) once without any repeats - a salesman's dream route. However, finding such a cycle is not an easy task, especially as the size of the graph grows, and this challenge is at the heart of what makes the TSP so computationally dense.

Polynomial Time Complexity

When we talk about solving problems like the Traveling Salesperson Problem, time is of the essence. Not actual time, but the amount of steps or operations it takes for a computer to solve the problem. This is what we call 'time complexity'.

In computer science, we often categorize problems based on the pattern of how their solving time grows, depending on the size of their inputs. If a problem can be solved in 'polynomial time', it means that the number of operations necessary to solve the problem can be expressed as a polynomial function of the size of the input. This is often denoted by 'big O' notation.

For instance, if a graph has 'E' edges, and it takes us to review each edge only once, then the problem can be solved in a time that grows proportionally to the number of edges. That's what we refer to as 'O(E)' - it's like saying the time it takes is related to the number of edges in a simple, direct way. The conversion in the TSP from an undirected to a directed graph being polynomial in time complexity means the problem remains manageable as the network of cities grows, offering a clue into the problem's solvable nature when utilizing effective algorithms.