computer science

Zadeh's rule

Zadeh's rule 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 Zadeh's rule rather than just read about it. In short: In mathematical optimization, Zadeh's rule (also known as the least-entered rule) is an algorithmic refinement of the simplex method for linear optimization. The rule was proposed around 1980 by Norman Zadeh (son of Lotfi A.

Key takeaways

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

Reference excerpt

In mathematical optimization, Zadeh's rule (also known as the least-entered rule) is an algorithmic refinement of the simplex method for linear optimization. The rule was proposed around 1980 by Norman Zadeh (son of Lotfi A. Zadeh), and has entered the folklore of convex optimization since then. Zadeh offered a reward of $1,000 to anyone who can show that the rule admits polynomially many iterations or to prove that there is a family of linear programs on which the pivoting rule requires subexponentially many iterations to find the optimum.

Algorithm Zadeh's rule belongs to the family of history-based improvement rules which, during a run of the simplex algorithm, retain supplementary data in addition to the current basis of the linear program. In particular, the rule chooses among all improving variables one which has entered the basis least often, intuitively ensuring that variables that might yield a substantive improvement in the long run but only a small improvement in a single step will be applied after a linear number of steps. The supplementary data structure in Zadeh's algorithm can therefore be modeled as an occurrence record, mapping all variables to natural numbers, monitoring how often a particular variable has entered the basis. In every iteration, the algorithm then selects an improving variable that is minimal with respect to the retained occurrence record. Note that the rule does not explicitly specify which particular improving variable should enter the basis in case of a tie.

Superpolynomial lower bound Zadeh's rule has been shown to have at least super-polynomial time complexity in the worse-case by constructing a family of Markov decision processes on which the policy iteration algorithm requires a super-polynomial number of steps. Running the simplex algorithm with Zadeh's rule on the induced linear program then yields a super-polynomial lower bound. The result was presented at the "Efficiency of the Simplex Method: Quo vadis Hirsch conjecture?" IPAM workshop in 2011 by Oliver Friedmann. Zadeh, although not working in academia anymore at that time, attended the Workshop and honored his original proposal.

Exponential lower bound Friedmann's original result has since been strengthened by the construction of an exponential instance for Zadeh's rule.

Notes

Worked examples

Example 1 — a first encounter with Zadeh's rule

Start with the simplest possible case. Write down what Zadeh's rule 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 Zadeh's rule 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 Zadeh's rule 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 Zadeh's rule

In research: Zadeh's rule 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 Zadeh's rule 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: Zadeh's rule is common in secondary-school and first-year university syllabi. It links to neighbouring topics Exchange algorithms, Linear programming, Optimization algorithms and methods, so understanding it makes those chapters shorter.
In everyday life: Look for Zadeh's rule 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 Zadeh's rule in 20 minutes

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

Frequently asked questions

What is Zadeh's rule in simple terms?

Why does Zadeh's rule 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 Zadeh's rule?

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 Zadeh's rule.