computer science

Mixed Chinese postman problem

Mixed Chinese postman problem is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Mixed Chinese postman problem rather than just read about it. In short: The mixed Chinese postman problem (MCPP or MCP) is the search for the shortest traversal of a graph with a set of vertices V, a set of undirected edges E with positive rational weights, and a set of directed arcs A with positive rational weights that covers each edge or arc at least once at minimal cost. The problem has been proven to be NP-complete by Papadimitriou.

Key takeaways

Mixed Chinese postman problem belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Mixed Chinese postman problem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Mixed Chinese postman problem from memory before moving on to harder problems.

Reference excerpt

The mixed Chinese postman problem (MCPP or MCP) is the search for the shortest traversal of a graph with a set of vertices V, a set of undirected edges E with positive rational weights, and a set of directed arcs A with positive rational weights that covers each edge or arc at least once at minimal cost. The problem has been proven to be NP-complete by Papadimitriou. The mixed Chinese postman problem often arises in arc routing problems such as snow ploughing, where some streets are too narrow to traverse in both directions while other streets are bidirectional and can be plowed in both directions. It is easy to check if a mixed graph has a postman tour of any size by verifying if the graph is strongly connected. The problem is NP hard if we restrict the postman tour to traverse each arc exactly once or if we restrict it to traverse each edge exactly once, as proved by Zaragoza Martinez. The problem is a more complex version of the Chinese postman problem.

Mathematical Definition The mathematical definition is: Input: A strongly connected, mixed graph G = ( V , E , A ) {\displaystyle G=(V,E,A)} with cost c ( e ) ≥ 0 {\displaystyle c(e)\geq 0} for every edge e ⊂ E ∪ A {\displaystyle e\subset E\cup A} and a maximum cost c m a x {\displaystyle c_{max}} . Question: is there a (directed) tour that traverses every edge in E {\displaystyle E} and every arc in A {\displaystyle A} at least once and has cost at most c m a x {\displaystyle c_{max}} ?

Computational complexity The main difficulty in solving the Mixed Chinese Postman problem lies in choosing orientations for the (undirected) edges when we are given a tight budget for our tour and can only afford to traverse each edge once. We then have to orient the edges and add some further arcs in order to obtain a directed Eulerian graph, that is, to make every vertex balanced. If there are multiple edges incident to one vertex, it is not an easy task to determine the correct orientation of each edge. The mathematician Papadimitriou analyzed this problem with more restrictions; "MIXED CHINESE POSTMAN is NP-complete, even if the input graph is planar, each vertex has degree at most three, and each edge and arc has cost one."

Eulerian graph The process of checking if a mixed graph is Eulerian is important to creating an algorithm to solve the Mixed Chinese Postman problem. The degrees of a mixed graph G must be even to have an Eulerian cycle, but this is not sufficient.

Approximation The fact that the Mixed Chinese Postman is NP-hard has led to the search for polynomial time algorithms that approach the optimum solution to reasonable threshold. Frederickson developed a method with a factor of 3/2 that could be applied to planar graphs, and Raghavachari and Veerasamy found a method that does not have to be planar. However, polynomial time cannot find the cost of deadheading, the time it takes a snow plough to reach the streets it will plow or a street sweeper to reach the streets it will sweep.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Mixed Chinese postman problem

Start with the simplest possible case. Write down what Mixed Chinese postman problem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Mixed Chinese postman problem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Mixed Chinese postman problem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Mixed Chinese postman problem

In research: Mixed Chinese postman problem appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Mixed Chinese postman problem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Mixed Chinese postman problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational problems in graph theory, so understanding it makes those chapters shorter.
In everyday life: Look for Mixed Chinese postman problem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Mixed Chinese postman problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Mixed Chinese postman problem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Mixed Chinese postman problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Mixed Chinese postman problem in simple terms?

Why does Mixed Chinese postman problem matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Mixed Chinese postman problem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Mixed Chinese postman problem.