Question

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

Short answer

Yes, the Hamiltonian Circuits Decision problem can be reduced to the Traveling Salesperson Decision problem in polynomial time. This is done by setting the weight of each edge to 1 and setting the budget B equal to the number of nodes.

Step-by-step solution

Understanding the Problems Involved

Firstly, make sure to understand both the Hamiltonian circuit decision problem and the Traveling Salesperson problem. A Hamiltonian circuit is a closed loop on a graph where every node (vertex) is visited exactly once. A Traveling Salesperson problem is an optimization problem that aims to determine the shortest possible route that a traveling salesperson can take to visit each city once and return to the original city. The version we are dealing with here is a decision problem, where the question is whether such a tour exists within a specified budget B.

Defining The Reduction

Now describe a method for converting any instance of the Hamiltonian Circuit Decision problem to an instance of the Traveling Salesperson Decision problem, such that the former has a Hamiltonian circuit if and only if the latter has a tour of cost at most B. As the weights of the edges are not part of the Hamiltonian Circuit problem, assign a weight of 1 to every edge in the graph. Then, for the Traveling Salesman problem, set the budget B equal to the number of nodes in the graph, which is the number of edges in the Hamiltonian circuit.

Proving The Reduction

Finally, the Hamiltonian circuit decision problem is reducible to the Traveling Salesperson Decision problem if an algorithm exists that performs this reduction in polynomial time. The reduction described simply involves setting the weight of each edge to 1 and setting the budget B equal to the number of nodes, which can be done in linear time with respect to the number of edges. Therefore, this is a polynomial time reduction.

Key concepts

Hamiltonian Circuit

A Hamiltonian circuit is a fundamental concept in graph theory. It refers to a closed loop that travels through each vertex of a graph exactly once before returning to the starting vertex. This tour encompasses all vertices without repetition, making it both unique and challenging to determine in complex graphs.
One important aspect of the Hamiltonian circuit is that it demands planning to ensure each vertex is visited precisely once. As opposed to just any path or journey, a Hamiltonian circuit covers all points of interest efficiently.

• **Key Points:**
• All vertices are visited exactly once.
• The circuit returns to the starting vertex.
• Existence in a graph is not guaranteed and can be difficult to ascertain.

Traveling Salesperson Problem

The Traveling Salesperson Problem (TSP) is a classic optimization problem. It asks for the shortest possible route that allows a salesperson to visit each city exactly once and return to the starting point. TSP is a fundamental example of a class of problems in computational complexity known as NP-hard.
In TSP, the challenge is not just finding a route but finding the most efficient one in terms of travel cost or time. When approached as a decision problem, the essential question becomes whether such a route exists within a given budget.

• **Challenges and Considerations in TSP:**
• Decision problem form asks if there is a route within a specific budget.
• Routes must optimize cost/time.
• Simplicity in smaller cities, complexity rises with city number.

Decision Problem

A decision problem focuses on answering a yes-or-no question based on specific constraints or criteria. It transforms the typical optimization problem into a more straightforward question regarding feasibility.
In the context of the Hamiltonian circuit and the TSP, the decision problem asks if there exists a tour (Hamiltonian circuit or TSP route) that meets certain constraints, such as budget or visiting all vertices.

• **Noteworthy Elements:**
• Transforms complex problems into simpler questions.
• Reduction from one problem to another often involves crafting a decision problem.
• Helps in determining feasibility rather than the optimal solution.

Graph Theory

Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. It is widely applicable in areas ranging from computer science to biology, offering a framework for problem-solving in many disciplines.
In graph theory, a graph consists of nodes (or vertices) and edges connecting these nodes. Many problems such as the Hamiltonian Circuit and TSP have their roots in understanding the properties and behaviors of these graphs.

• **Graph Theory Essentials Include:**
• Nodes (vertices) and connections (edges).
• Application in network analysis, scheduling, and more.
• Fundamental in solving problems like Hamiltonian circuits and TSP.