# Approximation algorithm

In

computer science andoperations research ,**approximation algorithms**arealgorithm s used to find approximate solutions tooptimization problem s. Approximation algorithms are often associated withNP-hard problems; since it is unlikely that there can ever be efficientpolynomial 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 (for instance within 5% of the optimal solution). 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 (such a family of approximation algorithms is often called apolynomial time approximation scheme or "PTAS"). Others are impossible to approximate within any constant, or even polynomial factor unlessP = NP , such as themaximum clique problem .NP-hard problems can often be expressed as

integer programs (IP) and solved exactly inexponential time . Many approximation algorithms emerge from thelinear programming relaxation of the integer program.Not all approximation algorithms are suitable for all practical applications. They often use IP/LP/Semidefinite solvers, complex data structures or sophisticated algorithmic techniques which lead to difficult implementation problems. Also, some approximation algorithms have impractical running times even though they are polynomial time, for example O("n"

^{2000}). Yet the study of even very expensive algorithms is not a completely theoretical pursuit as they can yield valuable insights. A classic example is the initial PTAS for Euclidean TSP due toSanjeev Arora which had prohibitive running time, yet within a year, Arora refined the ideas into a linear time algorithm. Such algorithms are also worthwhile in some applications where the running times and cost can be justified e.g. computational biology, financial engineering, transportation planning, and inventory management. In such scenarios, they must compete with the corresponding direct IP formulations.Another limitation of the approach is that it applies only to optimization problems and not to "pure"

decision problem s like satisfiability, although it is often possible to conceive optimization versions of such problems, such as themaximum satisfiability problem .Inapproximability has been a fruitful area of research in computational complexity theory since the 1990 result of Feige, Goldwasser, Lovasz, Safra and Szegedy on the inapproximability of Independent Set. After Arora et al. proved the PCP theorem a year later, it has now been shown that Johnson's 1974 approximation algorithms for Max SAT, Set Cover, Independent Set and Coloring all achieve the optimal approximation ratio, assuming P != NP.

**Performance guarantees**For some approximation algorithms it is possible to prove certain properties about the approximation of the optimum result. For example, in the case of a

**"ρ"-approximation algorithm**it has been proven that the approximation "a" will not be more (or less, depending on the situation) than a factor "ρ" times the optimum solution "s".:$egin\{cases\}s\; leq\; a\; leq\; ho\; s,qquadmbox\{if\; \}\; ho\; 1;\; \backslash \; ho\; s\; leq\; a\; leq\; s,qquadmbox\{if\; \}\; ho\; 1.end\{cases\}$

The factor "ρ" is called the "relative performance guarantee". An approximation algorithm has an "absolute performance guarantee" or "bounded error" "ε", if it has been proven that

:$(s\; -\; epsilon)\; leq\; a\; leq\; (s\; +\; epsilon).$

Similarly, the "absolute performance ratio" $Rho\_A$ of some approximation algorithm $A$, where $I$ refers to an instance of a problem, and where $R\_A(I)$ is the performance guarantee of $A$ on $I$ (i.e. $ho$ for problem instance $I$) is:

:$Rho\_A\; =\; inf\; \{\; r\; geq\; 1\; |\; R\_A(I)\; leq\; r,\; forall\; I\; \}$

That is to say that $Rho\_A$ is the largest bound on the approximation ratio, $r$, that one sees over all possible instances of the problem. Likewise, the "asymptotic performance ratio" $R\_A^infty$ is:

:$R\_A^infty\; =\; inf\; \{\; r\; geq\; 1\; |\; exists\; n\; in\; mathbb\{Z\}^+,\; R\_A(I)\; leq\; r,\; forall\; I,\; I\; geq\; n\}$

That is to say that it is the same as the "absolute performance ratio", with a lower bound $n$ on the size of problem instances. These two types of ratios are used because there exist algorithms where the difference between these two is significant.

Domination analysis provides an alternative way to analyze the quality of an approximation algorithm in terms of the rank of the computed solution in the sorted sequence of all possible solutions.**References*** cite book

last = Vazirani

first = Vijay V.

authorlink = Vijay Vazirani

title = Approximation Algorithms

publisher = Springer

date = 2003

location = Berlin

isbn = 3540653678

*Thomas H. Cormen ,Charles E. Leiserson ,Ronald L. Rivest , andClifford Stein . "Introduction to Algorithms ", Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Chapter 35: Approximation Algorithms, pp.1022–1056.

*Dorit H. Hochbaum , ed. "Approximation Algorithms for NP-Hard problems , PWS Publishing Company, 1997. ISBN 0-534-94968-1. Chapter 9: Various Notions of Approximations: Good, Better, Best, and More**External links***Pierluigi Crescenzi, Viggo Kann, Magnús Halldórsson,

Marek Karpinski and Gerhard Woeginger, [*http://www.nada.kth.se/~viggo/wwwcompendium/ "A compendium of NP optimization problems"*] .

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### Look at other dictionaries:

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