science

Polynomial-time reduction

Polynomial-time reduction is a 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 Polynomial-time reduction rather than just read about it. In short: In computational complexity theory, a polynomial-time reduction is a method for solving one problem using another. One shows that if a hypothetical subroutine solving the second problem exists, then the first problem can be solved by transforming or reducing it to inputs for the second problem and calling the subroutine one or more times.

Key takeaways

Polynomial-time reduction belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Polynomial-time reduction to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Polynomial-time reduction from memory before moving on to harder problems.

Reference excerpt

In computational complexity theory, a polynomial-time reduction is a method for solving one problem using another. One shows that if a hypothetical subroutine solving the second problem exists, then the first problem can be solved by transforming or reducing it to inputs for the second problem and calling the subroutine one or more times. If both the time required to transform the first problem to the second and the number of times the subroutine is called are polynomial, then the first problem is polynomial-time reducible to the second. A polynomial-time reduction proves that the first problem is no more difficult than the second one, because whenever an efficient algorithm exists for the second problem, one exists for the first problem as well. By contraposition, if no efficient algorithm exists for the first problem, none exists for the second either. Polynomial-time reductions are frequently used in complexity theory for defining both complexity classes and complete problems for those classes.

Types of reductions The three most common types of polynomial-time reduction, from the most to the least restrictive, are polynomial-time many-one reductions, truth-table reductions, and Turing reductions. The most frequently used of these are the many-one reductions, and in some cases the phrase "polynomial-time reduction" may be used to mean a polynomial-time many-one reduction. The most general reductions are the Turing reductions and the most restrictive are the many-one reductions with truth-table reductions occupying the space in between.

Many-one reductions A polynomial-time many-one reduction from a problem A to a problem B (both of which are usually required to be decision problems) is a polynomial-time algorithm for transforming inputs to problem A into inputs to problem B, such that the transformed problem has the same output as the original problem. An instance x of problem A can be solved by applying this transformation to produce an instance y of problem B, giving y as the input to an algorithm for problem B, and returning its output. Polynomial-time many-one reductions may also be known as polynomial transformations or Karp reductions, named after Richard Karp. A reduction of this type is denoted by A ≤ m P B {\displaystyle A\leq _{m}^{P}B} or A ≤ p B {\displaystyle A\leq _{p}B} .

Truth-table reductions A polynomial-time truth-table reduction from a problem A to a problem B (both decision problems) is a polynomial time algorithm for transforming inputs to problem A into a fixed number of inputs to problem B, such that the output for the original problem can be expressed as a function of the outputs for B. The function that maps outputs for B into the output for A must be the same for all inputs, so that it can be expressed by a truth table. A reduction of this type may be denoted by the expression A ≤ t t P B {\displaystyle A\leq _{tt}^{P}B} .

Turing reductions A polynomial-time Turing reduction from a problem A to a problem B is an algorithm that solves problem A using a polynomial number of calls to a subroutine for problem B, and polynomial time outside of those subroutine calls. Polynomial-time Turing reductions are also known as Cook reductions, named after Stephen Cook. A reduction of this type may be denoted by the expression A ≤ T P B {\displaystyle A\leq _{T}^{P}B} . Many-one reductions can be regarded as restricted variants of Turing reductions where the number of calls made to the subroutine for problem B is exactly one and the value returned by the reduction is the same value as the one returned by the subroutine.

Completeness A complete problem for a given complexity class C and reduction ≤ is a problem P that belongs to C, such that every problem A in C has a reduction A ≤ P. For instance, a problem is NP-complete if it belongs to NP and all problems in NP have polynomial-time many-one reductions to it. A problem that belongs to NP can be proven to be NP-complete by finding a single polynomial-time many-one reduction to it from a known NP-complete problem. Polynomial-time many-one reductions have been used to define complete problems for other complexity classes, including the PSPACE-complete languages and EXPTIME-complete languages. Every decision problem in P (the class of polynomial-time decision problems) may be reduced to every other nontrivial decision problem (where nontrivial means that not every input has the same output), by a polynomial-time many-one reduction. To transform an instance of problem A to B, solve A in polynomial time, and then use the solution to choose one of two instances of problem B with different answers. Therefore, for complexity classes within P such as L, NL, NC, and P itself, polynomial-time reductions cannot be used to define complete languages: if they were used in this way, every nontrivial problem in P would be complete. Instead, weaker reductions such as log-space reductions or NC reductions are used for defining classes of complete problems for these classes, such as the P-complete problems.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Polynomial-time reduction

Start with the simplest possible case. Write down what Polynomial-time reduction claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In 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 Polynomial-time reduction 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 Polynomial-time reduction 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 Polynomial-time reduction

In research: Polynomial-time reduction appears in 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 Polynomial-time reduction 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: Polynomial-time reduction is common in secondary-school and first-year university syllabi. It links to neighbouring topics Reduction (complexity), so understanding it makes those chapters shorter.
In everyday life: Look for Polynomial-time reduction 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 Polynomial-time reduction in 20 minutes

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

Frequently asked questions

What is Polynomial-time reduction in simple terms?

Why does Polynomial-time reduction matter?

Because it connects several 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 Polynomial-time reduction?

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 Polynomial-time reduction.