Maximum cut

Key takeaways

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

Reference excerpt

In a graph, a maximum cut is a cut whose size is at least the size of any other cut. That is, it is a partition of the graph's vertices into two complementary sets S and T, such that the number of edges between S and T is as large as possible. Finding such a cut is known as the max-cut problem. The problem can be stated simply as follows. One wants a subset S of the vertex set such that the number of edges between S and the complementary subset is as large as possible. Equivalently, one wants a bipartite subgraph of the graph with as many edges as possible. There is a more general version of the problem called weighted max-cut, where each edge is associated with a real number, its weight, and the objective is to maximize the total weight of the edges between S and its complement rather than the number of the edges. The weighted max-cut problem allowing both positive and negative weights can be trivially transformed into a weighted minimum cut problem by flipping the sign in all weights.

Lower bounds Edwards obtained the following two lower bounds for maximum cuts on a graph G with n vertices and m edges:

For arbitrary graphs, the maximum cut is at least ⌈ m 2 + m 8 + 1 64 − 1 8 ⌉ . {\displaystyle \displaystyle \left\lceil {\frac {m}{2}}+{\sqrt {{\frac {m}{8}}+{\frac {1}{64}}}}-{\frac {1}{8}}\right\rceil .}

For connected graphs, it is at least m 2 + n − 1 4 . {\displaystyle \displaystyle {\frac {m}{2}}+{\frac {n-1}{4}}.}

The bound for connected graphs is often called the Edwards–Erdős bound as Erdős conjectured it. Edwards proved the Edwards-Erdős bound using the probabilistic method; Crowston et al. proved the bound using linear algebra and analysis of pseudo-boolean functions. The Edwards-Erdős bound extends to the Balanced Subgraph Problem (BSP) on signed graphs G = (V, E, s), i.e. graphs where each edge is assigned + or –. For a partition of V into subsets U and W, an edge xy is balanced if either s(xy) = + and x and y are in the same subset, or s(xy) = – and x and y are different subsets. BSP aims at finding a partition with the maximum number b(G) of balanced edges in G. The Edwards-Erdős gives a lower bound on b(G) for every connected signed graph G. Edwards's bound for arbitrary graphs was improved for special classes of graphs: triangle-free graphs, graphs of given maximum degree, H-free graphs, etc. Poljak and Turzik extended the Edwards-Erdős bound to weighted maximum cuts: the weight of a maximum cut is at least

w ( G ) 2 + w ( T m i n ) 4 , {\displaystyle {\frac {w(G)}{2}}+{\frac {w(T_{min})}{4}},} where w(G) and w(Tmin) are the weights of G and its minimum weight spanning tree Tmin. Gutin and Yeo obtained a number of lower bounds for weighted Max-Cut extending the Poljak-Turzik bound for arbitrary weighted graphs and bounds for special classes of weighted graphs.

Computational complexity The following decision problem related to maximum cuts has been studied widely in theoretical computer science:

This problem is known to be NP-complete. It is easy to see that the problem is in NP: a yes answer is easy to prove by presenting a large enough cut. The NP-completeness of the problem can be shown, for example, by a reduction from maximum 2-satisfiability (a restriction of the maximum satisfiability problem). The weighted version of the decision problem was one of Karp's 21 NP-complete problems; Karp showed the NP-completeness by a reduction from the partition problem. The canonical optimization variant of the above decision problem is usually known as the Maximum-Cut Problem or Max-Cut and is defined as:

The optimization variant is known to be NP-Hard. The opposite problem, that of finding a minimum cut is known to be efficiently solvable via the Ford–Fulkerson algorithm.

Algorithms

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Maximum cut

Start with the simplest possible case. Write down what Maximum 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 Maximum 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 Maximum 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 Maximum cut

In research: Maximum 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 Maximum 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: Maximum cut is common in secondary-school and first-year university syllabi. It links to neighbouring topics Combinatorial optimization, Computational problems in graph theory, Graph theory objects, so understanding it makes those chapters shorter.
In everyday life: Look for Maximum 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 Maximum cut in 20 minutes

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

Frequently asked questions

What is Maximum cut in simple terms?

In a graph, a maximum cut is a cut whose size is at least the size of any other cut. That is, it is a partition of the graph's vertices into two complementary sets S and T, such that the number of edges between S and T is as large as possible.

Why does Maximum 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 Maximum 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 Maximum cut.