## What do you mean by polynomial-time reduction?

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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.

**What is meant by polynomial time?**

An algorithm is said to be of polynomial time if its running time is upper bounded by a polynomial expression in the size of the input for the algorithm, that is, T(n) = O(nk) for some positive constant k.

**What is the purpose of reduce polynomial?**

(A) Reductions allow us to formalize the notion of “Problem X is no harder to solve than Problem Y ”. (B) If Problem X reduces to Problem Y (we write X ≤ Y ), then X cannot be harder to solve than Y .

### Is polynomial time efficient?

Polynomial time algorithms are considered efficient only in comparison with the hardest non-polynomial time especially the so called NP-Complete. See image: Euler diagram for P, NP, NP-complete, and NP-hard set of problems.

**What is Cook reduction?**

(definition) Definition: A reduction computed by a deterministic polynomial time oracle Turing machine. See also NP-complete, Turing reduction, Karp reduction, l-reduction, many-one reduction, polynomial-time reduction.

**What is polynomial time and exponential time?**

Polynomial time. A polynomial is a sum of terms that look like Constant * x^k Exponential means something like Constant * k^x. (in both cases, k is a constant and x is a variable). The execution time of exponential algorithms grows much faster than that of polynomial ones.

#### What is a polynomial function is?

A polynomial function is a function that involves only non-negative integer powers or only positive integer exponents of a variable in an equation like the quadratic equation, cubic equation, etc. For example, 2x+5 is a polynomial that has exponent equal to 1.

**What is polynomial time algorithm example?**

Many widely used algorithms have polynomial time complexity (like our algorithms readNumbers1 and readNumbers2 , quicksort, insertion sort, binary search etc. etc.). Examples of algorithms with non-polynomial time complexity are all kinds of brute-force algorithms that look through all possible configurations.

**What is Reducibility explain with example?**

Reducibility for any problem (NP-hard or any other) means the possibility to convert problem A into other problem B. If we know the complexity of problem B then the complexity of problem A is at least the same as the complexity of problem A.

## What are NP-hard problems explain the polynomial-time reduction with an example?

A simple example of an NP-hard problem is the subset sum problem. A more precise specification is: a problem H is NP-hard when every problem L in NP can be reduced in polynomial time to H; that is, assuming a solution for H takes 1 unit time, H’s solution can be used to solve L in polynomial time.

**Why do we care about polynomial time?**

Polynomials have nice closure properties. In particular, performing a fixed number of polynomial time operations will always finish in polynomial time. Anything that takes polynomial time on a Turing machine also takes polynomial time on all of the other Turing complete models of computation we know about.