Chinese Postman Problem (CPP): Complete Guide with Solutions 2024

Understanding the Chinese Postman Problem (CPP)

Origin and Background

The Chinese Postman Problem (CPP) is a kind of routing problem in which a postman must determine the quickest path to carry mail to every address on a certain set of streets. The issue is made more difficult by the possibility that some streets must be traversed more than once to finish the delivery route.

The goal of CPP is to find a route that reaches every street (or edge) at least once and returns to the starting location while reducing the overall distance traveled. It was first introduced by a Chinese mathematician named Kwan Mei-Ko in 1962 while studying a problem related to postal delivery in Beijing. This is the reason it is referred to as the “Chinese Postman Problem.”

Real-World Applications

There is no known algorithm that can solve the problems of vehicle routing in polynomial time for all instances because it is known to be NP-hard. This problem is useful in several fields including circuit design, DNA sequencing, and transportation planning.

Key Variables

The key variables in the Chinese Postman Problem include the number of edges (streets), their lengths, and whether they must be traversed once or multiple times. These factors influence the overall complexity of the problem and the optimization approach needed to minimize travel distance.

Transportation Planning

The Chinese Postman Problem’s practical relevance lies in its adaptability to transportation planning scenarios. It provides insights into route optimization and network analysis, offering foundational strategies for designing efficient systems and improving logistical operations.

Technical Components of CPP

Graph Theory Fundamentals

Graph theory studies the properties and applications of graphs that consist of vertices (or nodes) connected by edges (or links). 

Here are the basic components.

• Vertices: These points represent entities or objects in a connected graph. 
• Edges: These represent connections between vertices, and can be unidirectional or bidirectional. 
• Degree of a Vertex: It is the total number of edges that connect to a vertex.
• Types of Graphs: There are different graph forms, including: 

• Weighted graphs, where edges have associated costs 
• Bipartite graphs, where there are two distinct sets of vertices
• Eulerian/Hamiltonian graphs, which relate to specific path properties. 

A graph G is formally defined as G = (V, E), where V represents vertices and E represents edges.

Mathematical Formulation

The Chinese Postman Problem (CPP) focuses on determining the shortest closed path that you need to reach every edge of a graph at least once. The mathematical formulation can be expressed as follows:

• Graph Representation: Establish the problem using G = (V, E), where V represents vertices and locations, and E represents edges.
• Objective Function: Reduce the total distance traveled while covering all the edges at least once. 

Minimize  ⅀dexe

• Constraints: Ensure you traverse each edge at least once and form a closed loop. There may be limits on vertex visits based on logistics requirements.                  

Logistics Use Cases

In different cases that need efficient route planning, you can apply CPP. 

Below are the two examples. 

• Mail Delivery: This involves creating optimal routes for postal services so delivery personnel can cover all the streets while ensuring minimal travel distance.
• Garbage Collection: It entails optimizing collection or pickup routes to cover all sites most efficiently.

Transportation Planning

The CPP helps develop strategies to ensure the efficient movement of products or people across networks.

• Public Transit Systems: This helps ensure buses or trains reach all the stops while keeping operational costs at a minimum. 
• Emergency Response Routes: It helps plan optimal routes for ambulances or fire trucks to help reach the location in the shortest time during a crisis.

Complexity of Chinese Postman Problem

The Chinese Postman Problem (CPP) is also known as the Route Inspection Problem. It is a difficult problem to solve using conventional methods, which means it can take a very long time to identify the optimal solution. However, researchers have found that there are shortcuts that can be utilized to find the answer more quickly for specific kinds of graphs. 

Researchers have employed heuristic algorithms as one approach to solve the Chinese Postman Problem. Using approximation methods to create solutions that are close to the optimum solution is another way to handle the Chinese Postman Problem’s complexity. 

How Does Chinese Postman Problem (CPP) Work?

As we have learned the complexity of the Chinese Postman Problem, let us find out how it works:

Step 1: Determine the problem

The goal of the Chinese Postman Problem in graph theory involves finding the shortest path that passes by each edge of a given graph at least once. The objective is to make sure that all edges are covered as well as possible.

Step 2: Determine the odd vertices

We must first identify the odd vertices, or vertices with an odd number of edges, to solve CPP. This is crucial because the starting and ending sites must have an equal number of edges. 

Step 3: Pair the odd vertices 

After locating the odd vertices, we must couple them so that the total weight of the edges connecting them is as low as possible. This step makes sure that there are an equal amount of edges on each vertex.

Step 4: Create a new graph

The new graph that is produced next has both the original graph and the newly added edges from step three. The vertices of this new graph must all be of even degree.

Step 5: Find the Euler circuit 

Finally, if your input graph has the Euler circuit, it is a path that precisely traverses each graph edge. The CPP can be resolved by taking this route.

Even if the Chinese Postman Problem may appear to be challenging, the above steps guarantee that all graph edges are covered in the most effective way possible. 

Advanced Algorithm Implementation

The Chinese Postman Problem (CPP) can be solved using several different algorithms. Typical algorithms include the following:

1. Hierholzer’s algorithm explained

Implementation Steps:

1. Start with a graph that has an Eulerian circuit.
2. Select any vertex to begin the traversal.
3. Follow edges one at a time until returning to the starting vertex, creating a cycle.
4. Remove the repeated edges of the cycle from the graph.
5. Repeat the process for any remaining edges until all edges have been traversed.

Code Examples

def hierholzer(graph):

    stack = []

    circuit = []

    current_vertex = list(graph.keys())[0]

    while stack or graph[current_vertex]:

        if not graph[current_vertex]:

            circuit.append(current_vertex)

            current_vertex = stack.pop()

        else:

            stack.append(current_vertex)

            next_vertex = graph[current_vertex].pop()

            graph[next_vertex].remove(current_vertex)

            current_vertex = next_vertex

    circuit.append(current_vertex)

    return circuit

2. Christofides algorithm explained

This algorithm ensures that the final result is at most 1.5 times better than the weighted CPP’s ideal solution. The technique operates by first locating the graph’s lowest spanning tree, then locating a minimum weight match between vertices of odd degrees, and finally locating an Euler circuit that encompasses all of the graph’s edges.

Implementation Steps:

1. Compute the Minimum Spanning Tree (MST) of the graph.
2. Determine vertices with odd degrees in the MST.
3. Find a Minimum Weight Perfect Matching (MWPM) among these odd-degree vertices.
4. Combine the MST and MWPM to form a multigraph.
5. Find an Eulerian circuit in this multigraph.
6. Convert the Eulerian circuit into a Hamiltonian path by skipping visited nodes.

Code Examples

def christofides(graph):

    mst = minimum_spanning_tree(graph)

    odd_vertices = [v for v in mst if degree(mst, v) % 2 == 1]

    matching = minimum_weight_perfect_matching(odd_vertices, graph)

    eulerian_graph = combine(mst, matching)

    eulerian_circuit = find_eulerian_circuit(eulerian_graph)

    hamiltonian_path = convert_to_hamiltonian(eulerian_circuit)

    return hamiltonian_path

3. Blossom algorithm

A graph matching algorithm called the Blossom can be used to get the least weight match between vertices in the graph with odd degrees. This approach is built on the idea of augmenting pathways, which are paths that alternate between matched and unmatched edges while beginning and ending with unmatched vertices.

In general, every one of these algorithms offers a unique strategy for resolving the Chinese Postman Problem. 

Implementation Steps:

1. Initialize a matching as empty.
2. For each unmatched vertex, perform a search for augmenting paths.
3. Use alternating paths that switch between matched and unmatched edges to find augmenting paths.
4. Update the matching by adding the found augmenting path.

Code Examples

def blossom(graph):

    matching = {}

    def find_augmenting_path(start):

        # Implement BFS or DFS to find an augmenting path

        pass

    for vertex in graph:

        if vertex not in matching:

            path = find_augmenting_path(vertex)

            if path:

                # Update matching with the new augmenting path

                pass

    return matching

Common Misconceptions about CPP

Below are a few widespread misunderstandings about the Chinese Postman Problem, including:

• CPP only applies to mail delivery: Although the issue was first raised about mail delivery, it has now found use in a number of other fields as well.
• CPP can be solved efficiently for all instances: CPP is an NP-hard issue, as was already established, and there is currently no algorithm that can handle it effectively in all cases.
• CPP is only applicable to undirected graphs: The issue can be expanded to directed graphs and graphs with weighted edges even though it is commonly formulated for undirected graphs.
• CPP is only applicable to road networks: Although postal delivery was the initial inspiration for the issue, it has numerous real-world applications outside of road networks, including circuit design, travel planning, and DNA sequencing.
• CPP is only applicable to simple graphs: CPP can also be applied to more complex graphs such as multigraphs and hypergraphs.

To properly understand the variety of applications and intricacy of the Chinese Postman Problem, it is crucial to dispel these misconceptions. 

Conclusion

The Chinese Postman Problem (CPP) is a well-known problem in graph theory that has several real-world applications.  It is significant to remember that CPP applies to a variety of industries, including circuit board design and transportation, in addition to postal delivery. 

Furthermore, there are a number of widespread misconceptions about CPP, including the notion that it is effective in all situations and that it only applies to undirected graphs. The fundamentals of CPP and its applications can be helpful in many different industries, and computer science research is still being done in this area.