Definitions for approximation algorithm
ap·prox·i·ma·tion al·go·rithm

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1. approximation algorithmnoun

   A method of finding a nearly optimal solution to an optimization problem that cannot be solved exactly within a reasonable time.

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1. Approximation algorithm

   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. Approximation algorithms naturally arise in the field of theoretical computer science as a consequence of the widely believed P ≠ NP conjecture. Under this conjecture, a wide class of optimization problems cannot be solved exactly in polynomial time. The field of approximation algorithms, therefore, tries to understand how closely it is possible to approximate optimal solutions to such problems in polynomial time. In an overwhelming majority of the cases, the guarantee of such algorithms is a multiplicative one expressed as an approximation ratio or approximation factor i.e., the optimal solution is always guaranteed to be within a (predetermined) multiplicative factor of the returned solution. However, there are also many approximation algorithms that provide an additive guarantee on the quality of the returned solution. A notable example of an approximation algorithm that provides both is the classic approximation algorithm of Lenstra, Shmoys and Tardos for scheduling on unrelated parallel machines. The design and analysis of approximation algorithms crucially involves a mathematical proof certifying the quality of the returned solutions in the worst case. This distinguishes them from heuristics such as annealing or genetic algorithms, which find reasonably good solutions on some inputs, but provide no clear indication at the outset on when they may succeed or fail. There is widespread interest in theoretical computer science to better understand the limits to which we can approximate certain famous optimization problems. For example, one of the long-standing open questions in computer science is to determine whether there is an algorithm that outperforms the 1.5 approximation algorithm of Christofides to the metric traveling salesman problem. The desire to understand hard optimization problems from the perspective of approximability is motivated by the discovery of surprising mathematical connections and broadly applicable techniques to design algorithms for hard optimization problems. One well-known example of the former is the Goemans–Williamson algorithm for maximum cut, which solves a graph theoretic problem using high dimensional geometry.

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1. Approximation algorithm

   In computer science and operations research, approximation algorithms are algorithms used to find approximate solutions to optimization problems. Approximation algorithms are often associated with NP-hard problems; since it is unlikely that there can ever be efficient polynomial-time exact algorithms solving NP-hard problems, one settles for polynomial-time sub-optimal solutions. Unlike heuristics, which usually only find reasonably good solutions reasonably fast, one wants provable solution quality and provable run-time bounds. Ideally, the approximation is optimal up to a small constant factor. Approximation algorithms are increasingly being used for problems where exact polynomial-time algorithms are known but are too expensive due to the input size. A typical example for an approximation algorithm is the one for vertex cover in graphs: find an uncovered edge and add both endpoints to the vertex cover, until none remain. It is clear that the resulting cover is at most twice as large as the optimal one. This is a constant factor approximation algorithm with a factor of 2. NP-hard problems vary greatly in their approximability; some, such as the bin packing problem, can be approximated within any factor greater than 1. Others are impossible to approximate within any constant, or even polynomial factor unless P = NP, such as the maximum clique problem.2000

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Numerology

1. Chaldean Numerology

   The numerical value of approximation algorithm in Chaldean Numerology is: 3

2. Pythagorean Numerology

   The numerical value of approximation algorithm in Pythagorean Numerology is: 4

References

1. ^ Wiktionary
   https://en.wiktionary.org/wiki/Approximation_Algorithm
2. ^ Wikipedia
   https://en.wikipedia.org/wiki/Approximation_Algorithm
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=approximation algorithm