Minimum k-cut

Key takeaways

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

Reference excerpt

In mathematics, the minimum k-cut is a combinatorial optimization problem that requires finding a set of edges whose removal would partition the graph to at least k connected components. These edges are referred to as k-cut. The goal is to find the minimum-weight k-cut. This partitioning can have applications in VLSI design, data-mining, finite elements and communication in parallel computing.

Formal definition Given an undirected graph G = (V, E) with an assignment of weights to the edges w: E → N and an integer k ∈ { 2 , 3 , … , | V | } , {\displaystyle k\in \{2,3,\ldots ,|V|\},} partition V into k disjoint sets F = { C 1 , C 2 , … , C k } {\displaystyle F=\{C_{1},C_{2},\ldots ,C_{k}\}} while minimizing

∑ i = 1 k − 1 ∑ j = i + 1 k ∑ v 1 ∈ C i v 2 ∈ C j w ( { v 1 , v 2 } ) . {\displaystyle \sum _{i=1}^{k-1}\ \sum _{j=i+1}^{k}\sum _{\begin{smallmatrix}v_{1}\in C_{i}\\v_{2}\in C_{j}\end{smallmatrix}}w(\left\{v_{1},v_{2}\right\}).}

For a fixed k, the problem is polynomial time solvable in O ( | V | k 2 ) . {\displaystyle O{\bigl (}|V|^{k^{2}}{\bigr )}.} However, the problem is NP-complete if k is part of the input. It is also NP-complete if we specify k vertices and ask for the minimum k-cut which separates these vertices among each of the sets.

Approximations Several approximation algorithms exist with an approximation of 2 − 2 k . {\displaystyle 2-{\tfrac {2}{k}}.} A simple greedy algorithm that achieves this approximation factor computes a minimum cut in each of the connected components and removes the lightest one. This algorithm requires a total of n − 1 max flow computations. Another algorithm achieving the same guarantee uses the Gomory–Hu tree representation of minimum cuts. Constructing the Gomory–Hu tree requires n − 1 max flow computations, but the algorithm requires an overall O(kn) max flow computations. Yet, it is easier to analyze the approximation factor of the second algorithm. Moreover, under the small set expansion hypothesis (a conjecture closely related to the unique games conjecture), the problem is NP-hard to approximate to within (2 − ε) factor for every constant ε > 0, meaning that the aforementioned approximation algorithms are essentially tight for large k. A variant of the problem asks for a minimum weight k-cut where the output partitions have pre-specified sizes. This problem variant is approximable to within a factor of 3 for any fixed k if one restricts the graph to a metric space, meaning a complete graph that satisfies the triangle inequality. More recently, polynomial time approximation schemes (PTAS) were discovered for those problems. While the minimum k-cut problem is W[1]-hard parameterized by k, a parameterized approximation scheme can be obtained for this parameter.

Illustrations

Worked examples

Example 1 — a first encounter with Minimum k-cut

Start with the simplest possible case. Write down what Minimum k-cut 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 Minimum k-cut 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 Minimum k-cut 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 Minimum k-cut

In research: Minimum k-cut 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 Minimum k-cut 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: Minimum k-cut is common in secondary-school and first-year university syllabi. It links to neighbouring topics Approximation algorithms, Combinatorial optimization, Computational problems in graph theory, so understanding it makes those chapters shorter.
In everyday life: Look for Minimum k-cut 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.

How to study Minimum k-cut in 20 minutes

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

Frequently asked questions

What is Minimum k-cut in simple terms?

In mathematics, the minimum k-cut is a combinatorial optimization problem that requires finding a set of edges whose removal would partition the graph to at least k connected components. These edges are referred to as k-cut.

Why does Minimum k-cut 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 Minimum k-cut?

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 Minimum k-cut.