Question

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

Short answer

The reduction involves creating a weighted graph from the unweighted graph of the HCDP where each edge has weight 1. Set K in the TSDP as the total number of vertices in the graph. Therefore, for each Hamiltonian circuit in the HCDP, there exists a tour in the TSDP equivalent to that circuit with total weight as \(|V|\). The process only involves assigning weight and setting K which can be done in polynomial time.

Step-by-step solution

Describe the Hamiltonian Circuits Decision Problem

The Hamiltonian Circuit Decision Problem (HCDP) deals with a graph G(V,E) and inquires if there exists a Hamiltonian circuit in this graph. A Hamiltonian circuit is a closed loop on a graph where every vertex (V) is visited exactly once.

Describe the Traveling Salesperson Decision Problem

The Traveling Salesperson Decision Problem (TSP) deals with a complete graph G(V,E) with weighted edges. The question is whether there is a tour of this graph, visiting every vertex exactly once, such that the total weight of the tour is less than or equal to a given weight K.

Define the Reduction

A reduction from HCDP to TSDP can be formulated as follows: Take an instance of the HCDP as input, which is an unweighted graph G(V,E). Create a new weighted graph G'(V,E) where each edge has weight one (1). Set the given weight K for TSDP as the total number of vertices in the graph, which is \(|V|\). This essentially implies that for each Hamiltonian circuit in the HCDP, there exists a tour in the TSDP equivalent to that circuit with total weight as \(|V|\).

Prove the Reduction is Polynomial Time

In this case, the reduction only involves assigning a weight of 1 to each edge in the original graph and setting K to \(|V|\). Both these operations require scanning all vertices and edges once, and hence they can be done in polynomial time with respect to the size of the graph, proving our point.

Key concepts

Traveling Salesperson Decision Problem

The Traveling Salesperson Decision Problem (TSDP) is a classic question in computer science, particularly within the branch called combinatorial optimization. It presents a scenario where a salesperson must travel to a list of cities, visiting each city once before returning to the origin point. The challenge is to determine the shortest possible route that accomplishes this task.

Imagine you have a graph with cities as vertices and roads as edges. Each road (edge) has a cost (weight), and you want to find the least expensive path that visits every city once and returns to the starting city.

The TSDP is more than just a theoretical concept; it has real-world applications in logistics, manufacturing, and even DNA sequencing. The decision variant of the problem asks a yes-or-no question: Given a graph and a threshold cost K, is there a route that visits each city once with a total cost that doesn't exceed K? Despite its simple formulation, TSDP is notoriously difficult to solve for large numbers of cities, exemplifying problems that are easy to state but hard to solve.

Algorithmic Reduction

Algorithmic reduction is a technique used to show how one problem can be transformed into another, often revealing their computational similarities or difficulties. It's like translating a question from one language into another while retaining the complexity of the original problem.

In our case, we're looking at reducing the Hamiltonian Circuits Decision Problem (HCDP) to the Traveling Salesperson Decision Problem (TSDP). To accomplish this, we tweak the original problem slightly without changing its fundamental nature. For instance, by assigning a weight of 1 to each edge in an unweighted graph and limiting the acceptable total weight to the number of vertices, we are translating the requirement of visiting each vertex once into a context the TSDP can understand.

This process is crucial in the field of computational complexity. If we can reduce a well-known difficult problem to another, it suggests the second problem is at least as hard. This is how we often show that new problems belong to classes like NP-complete, which holds some of the most computationally challenging questions we know of.

Polynomial Time Complexity

Polynomial time complexity is a concept in computer science used to classify algorithms based on how the time to complete an algorithm increases with the size of the input data. An algorithm is said to run in polynomial time if its running time can be expressed as a polynomial function of the size of the input. Polynomial functions include linear (), quadratic (^2), cubic (^3), and so on, but not exponential functions like 2^n.

In simpler terms, if an algorithm's time to solve a problem grows at a rate that's proportional to a power of the size of the input, we say it has polynomial time complexity. These algorithms are considered efficient and feasible for practical use, especially when the data inputs are large.

The reduction from HCDP to TSDP can be done in polynomial time because it involves simple operations - assigning a constant weight to each edge and setting a limit equal to the number of vertices. These operations can be completed quickly, even as the graph grows, maintaining a practical time frame for solving even larger instances. This is a key point, as it indicates that the difficulty of solving the reduced problem (TSDP) is a reasonable, efficient step away from the original problem (HCDP).