How approximation algorithms are used for NP-hard problems?
In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems (in particular NP-hard problems) with provable guarantees on the distance of the returned solution to the optimal one.
What is NP-hard optimization problem?
A problem is NP-hard if an algorithm for solving it can be translated into one for solving any NP- problem (nondeterministic polynomial time) problem. NP-hard therefore means “at least as hard as any NP-problem,” although it might, in fact, be harder.
How do you prove an optimization problem is NP-hard?
A polynomial time algorithm to solve the optimization version of the problem would also solve the decision version in polynomial time (just compare the optimal value to the target value for the decision problem.) Thus if the decision problem is NP-Complete, than the optimization problem is NP-Hard.
What is NP-problem in algorithm?
A problem is called NP (nondeterministic polynomial) if its solution can be guessed and verified in polynomial time; nondeterministic means that no particular rule is followed to make the guess. If a problem is NP and all other NP problems are polynomial-time reducible to it, the problem is NP-complete.
Can optimization problems be NP?
Optimization problems can have related decision problems that are in NP, and these related decision problems can be NP-complete. For example, finding a minimum size vertex cover is an optimization problem, but determining whether there exists a vertex cover of size at most k is a decision problem.
How do you convert an optimization problem to a decision problem?
Note that the decision problem PATH is one way of casting the original optimization problem as a decision problem. We have done this by imposing a bound on the value to be optimized. This is a popular way of transforming an optimization problem into a decision problem.
What are approximation algorithms?
Approximation algorithms are efficient algorithms that find approximate solutions to optimization problems with provable guarantees on the distance of the returned solution to the optimal one. An optimization problem consists in finding the best (cheapest, heaviest, etc.) element in a large set P, called the feasible region.
Are there any polynomial-time optimization algorithms for NP-complete problems?
Their optimization versions fall in the class of NP-hard problems—problems that are at least as hard as NP-complete problems.2 Hence, there are no known polynomial-time algorithms for these problems, and there are serious theoretical reasons to believe that such algorithms do not exist.
What is an approximation ratio in Computer Science?
In the worst-case paradigm, algorithms for NP-hard problems are typically characterized by their approximation ratio, defined as the ratio between the worstcase cost of the algorithm and the cost of an all-powerful optimal algorithm. Many computationally-hard problems admit efficient worst-case approximations [28, 32,…
What is a polynomial-time approximation scheme?
A family of algorithms that can achieve any constant approximation error in polynomial time is called a polynomial-time approximation scheme or PTAS. To illustrate Approximation algorithms with small additive error, we have covered 3 problems:
