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Klee–Minty cube

Klee–Minty cube 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 Klee–Minty cube rather than just read about it. In short: The Klee–Minty cube or Klee–Minty polytope (named after Victor Klee and George J. Minty) is a unit hypercube of variable dimension whose corners have been perturbed.

Klee–Minty cube — main illustration
Klee–Minty cube — illustration

Key takeaways

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

Reference excerpt

The Klee–Minty cube or Klee–Minty polytope (named after Victor Klee and George J. Minty) is a unit hypercube of variable dimension whose corners have been perturbed. Klee and Minty demonstrated that George Dantzig's simplex algorithm has poor worst-case performance when initialized at one corner of their "squashed cube". On the three-dimensional version, the simplex algorithm and the criss-cross algorithm visit all 8 corners in the worst case. In particular, many optimization algorithms for linear optimization exhibit poor performance when applied to the Klee–Minty cube. In 1973 Klee and Minty showed that Dantzig's simplex algorithm was not a polynomial-time algorithm when applied to their cube. Later, modifications of the Klee–Minty cube have shown poor behavior both for other basis-exchange pivoting algorithms and also for interior-point algorithms.

Description The Klee–Minty cube was originally specified with a parameterized system of linear inequalities, with the dimension as the parameter. The cube in two-dimensional space is a squashed square, and the "cube" in three-dimensional space is a squashed cube. Illustrations of the "cube" have appeared besides algebraic descriptions. The Klee–Minty polytope is given by:

x 1 ≤ 5 4 x 1 + x 2 ≤ 25 8 x 1 + 4 x 2 + x 3 ≤ 125 ⋮ 2 D x 1 + 2 D − 1 x 2 + ⋯ + 4 x D − 1 + x D ≤ 5 D x 1 ≥ 0 , … , x D ≥ 0. {\displaystyle {\begin{aligned}x_{1}&\leq 5\\4x_{1}+x_{2}&\leq 25\\8x_{1}+4x_{2}+x_{3}&\leq 125\\&\vdots \\2^{D}x_{1}+2^{D-1}x_{2}+\dots +4x_{D-1}+x_{D}&\leq 5^{D}\\x_{1}\geq 0,\,\,\dots ,\,\,x_{D}&\geq 0.\end{aligned}}}

This has D {\displaystyle D} variables, D {\displaystyle D} constraints other than the D {\displaystyle D} non-negativity constraints, and 2 D {\displaystyle 2^{D}} vertices, just as a D {\displaystyle D} -dimensional hypercube does. If the objective function to be maximized is

2 D − 1 x 1 + 2 D − 2 x 2 + ⋯ + 2 x D − 1 + x D , {\displaystyle 2^{D-1}x_{1}+2^{D-2}x_{2}+\dots +2x_{D-1}+x_{D},}

and if the initial vertex for the simplex algorithm is the origin, then the algorithm as formulated by Dantzig visits all 2 D {\displaystyle 2^{D}} vertices, finally reaching the optimal vertex ( 0 , 0 , … , 5 D ) {\displaystyle (0,0,\dots ,5^{D})} .

… excerpt ends here. Continue reading the full article.

Illustrations

Klee–Minty cube: Klee Minty cube for shadow vertex simplex method.
Klee Minty cube for shadow vertex simplex method.
Klee–Minty cube: An illustration of a three-dimensional polytope which is the feasible region for a linear programming problem. The simplex algorithm traverses the edges between vertices until it reaches an optimal vertex. In the case shown, the simplex algorithm takes five steps. However, the simplex algorithm visits every vertex in the worst case of a problem whose feasible region is the Klee–Minty cube, so the number of steps rises exponentially with the dimension of the problem.
An illustration of a three-dimensional polytope which is the feasible region for a linear programming problem. The simplex algorithm traverses the edges between vertices until it reaches an optimal vertex. In the case shown, the simplex algorithm takes five steps. However, the simplex algorithm visits every vertex in the worst case of a problem whose feasible region is the Klee–Minty cube, so the number of steps rises exponentially with the dimension of the problem.

Worked examples

Example 1 — a first encounter with Klee–Minty cube

Start with the simplest possible case. Write down what Klee–Minty cube 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 Klee–Minty cube 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 Klee–Minty cube 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 Klee–Minty cube

In research
Klee–Minty cube 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 Klee–Minty cube 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
Klee–Minty cube is common in secondary-school and first-year university syllabi. It links to neighbouring topics Analysis of algorithms, Computational complexity theory, Convex geometry, so understanding it makes those chapters shorter.
In everyday life
Look for Klee–Minty cube 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 Klee–Minty cube in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Klee–Minty cube means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Klee–Minty cube out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Klee–Minty cube in simple terms?

The Klee–Minty cube or Klee–Minty polytope (named after Victor Klee and George J. Minty) is a unit hypercube of variable dimension whose corners have been perturbed.

Why does Klee–Minty cube 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 Klee–Minty cube?

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 Klee–Minty cube.

Tags

  • Analysis of algorithms
  • Computational complexity theory
  • Convex geometry
  • Cubes
  • Linear programming

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