computer science

Network flow problem

Network flow 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 Network flow problem rather than just read about it. In short: In combinatorial optimization, network flow problems are a class of computational problems in which the input is a flow network (a graph with numerical capacities on its edges), and the goal is to construct a flow, numerical values on each edge that respect the capacity constraints and that have incoming flow equal to outgoing flow at all vertices except for certain designated terminals. Specific types of network fl…

Key takeaways

Network flow 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 Network flow problem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Network flow problem from memory before moving on to harder problems.

Reference excerpt

In combinatorial optimization, network flow problems are a class of computational problems in which the input is a flow network (a graph with numerical capacities on its edges), and the goal is to construct a flow, numerical values on each edge that respect the capacity constraints and that have incoming flow equal to outgoing flow at all vertices except for certain designated terminals. Specific types of network flow problems include:

The maximum flow problem, in which the goal is to maximize the total amount of flow out of the source terminals and into the sink terminals The minimum-cost flow problem, in which the edges have costs as well as capacities and the goal is to achieve a given amount of flow (or a maximum flow) that has the minimum possible cost The multi-commodity flow problem, in which one must construct multiple flows for different commodities whose total flow amounts together respect the capacities Nowhere-zero flow, a type of flow studied in combinatorics in which the flow amounts are restricted to a finite set of nonzero values The max-flow min-cut theorem equates the value of a maximum flow to the value of a minimum cut, a partition of the vertices of the flow network that minimizes the total capacity of edges crossing from one side of the partition to the other. Approximate max-flow min-cut theorems provide an extension of this result to multi-commodity flow problems. The Gomory–Hu tree of an undirected flow network provides a concise representation of all minimum cuts between different pairs of terminal vertices. Algorithms for constructing flows include

Dinic's algorithm, a strongly polynomial algorithm for maximum flow The Edmonds–Karp algorithm, a faster strongly polynomial algorithm for maximum flow The Ford–Fulkerson algorithm, a greedy algorithm for maximum flow that is not in general strongly polynomial The network simplex algorithm, a method based on linear programming but specialized for network flow The out-of-kilter algorithm for minimum-cost flow The push–relabel maximum flow algorithm, one of the most efficient known techniques for maximum flow Otherwise the problem can be formulated as a more conventional linear program or similar and solved using a general purpose optimization solver.

References

Worked examples

Example 1 — a first encounter with Network flow problem

Start with the simplest possible case. Write down what Network flow 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 Network flow 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 Network flow 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 Network flow problem

In research: Network flow 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 Network flow 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: Network flow problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Combinatorial optimization, Directed graphs, Graph algorithms, so understanding it makes those chapters shorter.
In everyday life: Look for Network flow 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 Network flow problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Network flow 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 Network flow problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Network flow problem in simple terms?

Why does Network flow 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 Network flow 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 Network flow problem.