Definitions

# 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 (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 a polynomial time approximation scheme or PTAS). Others are impossible to approximate within any constant, or even polynomial factor unless P = NP, such as the maximum clique problem.

NP-hard problems can often be expressed as integer programs (IP) and solved exactly in exponential time. Many approximation algorithms emerge from the linear 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(n2000). 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 to Sanjeev 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 problems like satisfiability, although it is often possible to conceive optimization versions of such problems, such as the maximum 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.

$begin\left\{cases\right\}s leq a leq rho s,qquadmbox\left\{if \right\} rho > 1; rho s leq a leq s,qquadmbox\left\{if \right\} rho < 1.end\left\{cases\right\}$

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

$\left(s - epsilon\right) leq a leq \left(s + epsilon\right).$

Similarly, the absolute performance ratio $Rho_A$ of some approximation algorithm $A$, where $I$ refers to an instance of a problem, and where $R_A\left(I\right)$ is the performance guarantee of $A$ on $I$ (i.e. $rho$ for problem instance $I$) is:

$Rho_A = inf \left\{ r geq 1 | R_A\left(I\right) leq r, forall I \right\}$

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:

$R_A^infty = inf \left\{ r geq 1 | exists n in mathbb\left\{Z\right\}^+, R_A\left(I\right) leq r, forall I, I geq n\right\}$

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.

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