Approximation algorithm

A simple example of an approximation algorithm is one for the minimum vertex cover problem, where the goal is to choose the smallest set of vertices such that every edge in the input graph contains at least one chosen vertex. One way to find a vertex cover is to repeat the following process: find an uncovered edge, add both its endpoints to the cover, and remove all edges incident to either vertex from the graph. As any vertex cover of the input graph must use a distinct vertex to cover each edge that was considered in the process (since it forms a matching), the vertex cover produced, therefore, is at most twice as large as the optimal one. In other words, this is a constant-factor approximation algorithm with an approximation factor of 2. Under the recent unique games conjecture, this factor is even the best possible one.^[5]

NP-hard problems vary greatly in their approximability; some, such as the knapsack problem, can be approximated within a multiplicative factor ${\displaystyle 1+\epsilon }$, for any fixed ${\displaystyle \epsilon >0}$, and therefore produce solutions arbitrarily close to the optimum (such a family of approximation algorithms is called a polynomial-time approximation scheme or PTAS). Others are impossible to approximate within any constant, or even polynomial, factor unless P = NP, as in the case of the maximum clique problem. Therefore, an important benefit of studying approximation algorithms is a fine-grained classification of the difficulty of various NP-hard problems beyond the one afforded by the theory of NP-completeness. In other words, although NP-complete problems may be equivalent (under polynomial-time reductions) to each other from the perspective of exact solutions, the corresponding optimization problems behave very differently from the perspective of approximate solutions.

By now there are several established techniques to design approximation algorithms. These include the following ones.

1. Greedy algorithm
2. Local search
3. Enumeration and dynamic programming (which is also often used for parameterized approximations)
4. Solving a convex programming relaxation to get a fractional solution. Then converting this fractional solution into a feasible solution by some appropriate rounding. The popular relaxations include the following.
   • Linear programming relaxations
   • Semidefinite programming relaxations
5. Primal-dual methods
6. Dual fitting
7. Embedding the problem in some metric and then solving the problem on the metric. This is also known as metric embedding.
8. Random sampling and the use of randomness in general in conjunction with the methods above.

While approximation algorithms always provide an a priori worst case guarantee (be it additive or multiplicative), in some cases they also provide an a posteriori guarantee that is often much better. This is often the case for algorithms that work by solving a convex relaxation of the optimization problem on the given input. For example, there is a different approximation algorithm for minimum vertex cover that solves a linear programming relaxation to find a vertex cover that is at most twice the value of the relaxation. Since the value of the relaxation is never larger than the size of the optimal vertex cover, this yields another 2-approximation algorithm. While this is similar to the a priori guarantee of the previous approximation algorithm, the guarantee of the latter can be much better (indeed when the value of the LP relaxation is far from the size of the optimal vertex cover).

Approximation algorithms as a research area is closely related to and informed by inapproximability theory where the non-existence of efficient algorithms with certain approximation ratios is proved (conditioned on widely believed hypotheses such as the P ≠ NP conjecture) by means of reductions. In the case of the metric traveling salesman problem, the best known inapproximability result rules out algorithms with an approximation ratio less than 123/122 ≈ 1.008196 unless P = NP, Karpinski, Lampis, Schmied.^[6] Coupled with the knowledge of the existence of Christofides' 1.5 approximation algorithm, this tells us that the threshold of approximability for metric traveling salesman (if it exists) is somewhere between 123/122 and 1.5.

While inapproximability results have been proved since the 1970s, such results were obtained by ad hoc means and no systematic understanding was available at the time. It is only since the 1990 result of Feige, Goldwasser, Lovász, Safra and Szegedy on the inapproximability of Independent Set^[7] and the famous PCP theorem,^[8] that modern tools for proving inapproximability results were uncovered. The PCP theorem, for example, shows that Johnson's 1974 approximation algorithms for Max SAT, set cover, independent set and coloring all achieve the optimal approximation ratio, assuming P ≠ NP.^[9]

Not all approximation algorithms are suitable for direct practical applications. Some involve solving non-trivial linear programming/semidefinite relaxations (which may themselves invoke the ellipsoid algorithm), complex data structures, or sophisticated algorithmic techniques, leading to difficult implementation issues or improved running time performance (over exact algorithms) only on impractically large inputs. Implementation and running time issues aside, the guarantees provided by approximation algorithms may themselves not be strong enough to justify their consideration in practice. Despite their inability to be used "out of the box" in practical applications, the ideas and insights behind the design of such algorithms can often be incorporated in other ways in practical algorithms. In this way, the study of even very expensive algorithms is not a completely theoretical pursuit as they can yield valuable insights.

In other cases, even if the initial results are of purely theoretical interest, over time, with an improved understanding, the algorithms may be refined to become more practical. One such example is the initial PTAS for Euclidean TSP by Sanjeev Arora (and independently by Joseph Mitchell) which had a prohibitive running time of ${\displaystyle n^{O(1/\epsilon )}}$ for a ${\displaystyle 1+\epsilon }$ approximation.^[10] Yet, within a year these ideas were incorporated into a near-linear time ${\displaystyle O(n\log n)}$ algorithm for any constant ${\displaystyle \epsilon >0}$.^[11]

Given an optimization problem:

${\displaystyle \Pi :I\times S}$

where ${\displaystyle \Pi }$ is an approximation problem, ${\displaystyle I}$ the set of inputs and ${\displaystyle S}$ the set of solutions, we can define the cost function:

${\displaystyle c:S\rightarrow \mathbb {R} ^{+}}$

and the set of feasible solutions:

${\displaystyle \forall i\in I,S(i)={s\in S:i\Pi _{s}}}$

finding the best solution ${\displaystyle s^{*}}$ for a maximization or a minimization problem:

${\displaystyle s^{*}\in S(i)}$, ${\displaystyle c(s^{*})=min/max\ c(S(i))}$

Given a feasible solution ${\displaystyle s\in S(i)}$, with ${\displaystyle s\neq s^{*}}$, we would want a guarantee of the quality of the solution, which is a performance to be guaranteed (approximation factor).

Specifically, having ${\displaystyle A_{\Pi }(i)\in S_{i}}$, the algorithm has an approximation factor (or approximation ratio) of ${\displaystyle \rho (n)}$ if ${\displaystyle \forall i\in I\ s.t.|i|=n}$, we have:

• for a minimization problem: ${\displaystyle {\frac {c(A_{\Pi }(i))}{c(s^{*}(i))}}\leq \rho (n)}$, which in turn means the solution taken by the algorithm divided by the optimal solution achieves a ratio of ${\displaystyle \rho (n)}$;
• for a maximization problem: ${\displaystyle {\frac {c(s^{*}(i))}{c(A_{\Pi }(i))}}\leq \rho (n)}$, which in turn means the optimal solution divided by the solution taken by the algorithm achieves a ratio of ${\displaystyle \rho (n)}$;

The approximation can be proven tight (tight approximation) by demonstrating that there exist instances where the algorithm performs at the approximation limit, indicating the tightness of the bound. In this case, it's enough to construct an input instance designed to force the algorithm into a worst-case scenario.

For some approximation algorithms it is possible to prove certain properties about the approximation of the optimum result. For example, a ρ-approximation algorithm A is defined to be an algorithm for which it has been proven that the value/cost, f(x), of the approximate solution A(x) to an instance x will not be more (or less, depending on the situation) than a factor ρ times the value, OPT, of an optimum solution.

${\displaystyle {\begin{cases}\mathrm {OPT} \leq f(x)\leq \rho \mathrm {OPT} ,\qquad {\mbox{if }}\rho >1;\\\rho \mathrm {OPT} \leq f(x)\leq \mathrm {OPT} ,\qquad {\mbox{if }}\rho <1.\end{cases}}}$

The factor ρ is called the relative performance guarantee. An approximation algorithm has an absolute performance guarantee or bounded error c, if it has been proven for every instance x that

${\displaystyle (\mathrm {OPT} -c)\leq f(x)\leq (\mathrm {OPT} +c).}$

Similarly, the performance guarantee, R(x,y), of a solution y to an instance x is defined as

${\displaystyle R(x,y)=\max \left({\frac {OPT}{f(y)}},{\frac {f(y)}{OPT}}\right),}$

where f(y) is the value/cost of the solution y for the instance x. Clearly, the performance guarantee is greater than or equal to 1 and equal to 1 if and only if y is an optimal solution. If an algorithm A guarantees to return solutions with a performance guarantee of at most r(n), then A is said to be an r(n)-approximation algorithm and has an approximation ratio of r(n). Likewise, a problem with an r(n)-approximation algorithm is said to be r(n)-approximable or have an approximation ratio of r(n).^[12][13]

For minimization problems, the two different guarantees provide the same result and that for maximization problems, a relative performance guarantee of ρ is equivalent to a performance guarantee of ${\displaystyle r=\rho ^{-1}}$. In the literature, both definitions are common but it is clear which definition is used since, for maximization problems, as ρ ≤ 1 while r ≥ 1.

The absolute performance guarantee ${\displaystyle \mathrm {P} _{A}}$ of some approximation algorithm A, where x refers to an instance of a problem, and where ${\displaystyle R_{A}(x)}$ is the performance guarantee of A on x (i.e. ρ for problem instance x) is:

${\displaystyle \mathrm {P} _{A}=\inf\{r\geq 1\mid R_{A}(x)\leq r,\forall x\}.}$

That is to say that ${\displaystyle \mathrm {P} _{A}}$ is the largest bound on the approximation ratio, r, that one sees over all possible instances of the problem. Likewise, the asymptotic performance ratio ${\displaystyle R_{A}^{\infty }}$ is:

${\displaystyle R_{A}^{\infty }=\inf\{r\geq 1\mid \exists n\in \mathbb {Z} ^{+},R_{A}(x)\leq r,\forall x,|x|\geq n\}.}$

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.

Performance guarantees

	r-approx[12][13]	ρ-approx	rel. error[13]	rel. error[12]	norm. rel. error[12][13]	abs. error[12][13]
max: ${\displaystyle f(x)\geq }$	${\displaystyle r^{-1}\mathrm {OPT} }$	${\displaystyle \rho \mathrm {OPT} }$	${\displaystyle (1-c)\mathrm {OPT} }$	${\displaystyle (1-c)\mathrm {OPT} }$	${\displaystyle (1-c)\mathrm {OPT} +c\mathrm {WORST} }$	${\displaystyle \mathrm {OPT} -c}$
min: ${\displaystyle f(x)\leq }$	${\displaystyle r\mathrm {OPT} }$	${\displaystyle \rho \mathrm {OPT} }$	${\displaystyle (1+c)\mathrm {OPT} }$	${\displaystyle (1-c)^{-1}\mathrm {OPT} }$	${\displaystyle (1-c)^{-1}\mathrm {OPT} +c\mathrm {WORST} }$	${\displaystyle \mathrm {OPT} +c}$

In the literature, an approximation ratio for a maximization (minimization) problem of c - ϵ (min: c + ϵ) means that the algorithm has an approximation ratio of c ∓ ϵ for arbitrary ϵ > 0 but that the ratio has not (or cannot) be shown for ϵ = 0. An example of this is the optimal inapproximability — inexistence of approximation — ratio of 7 / 8 + ϵ for satisfiable MAX-3SAT instances due to Johan Håstad.^[14] As mentioned previously, when c = 1, the problem is said to have a polynomial-time approximation scheme.

An ϵ-term may appear when an approximation algorithm introduces a multiplicative error and a constant error while the minimum optimum of instances of size n goes to infinity as n does. In this case, the approximation ratio is c ∓ k / OPT = c ∓ o(1) for some constants c and k. Given arbitrary ϵ > 0, one can choose a large enough N such that the term k / OPT < ϵ for every n ≥ N. For every fixed ϵ, instances of size n < N can be solved by brute force, thereby showing an approximation ratio — existence of approximation algorithms with a guarantee — of c ∓ ϵ for every ϵ > 0.

• Domination analysis considers guarantees in terms of the rank of the computed solution.
• PTAS - a type of approximation algorithm that takes the approximation ratio as a parameter
• Parameterized approximation algorithm - a type of approximation algorithm that runs in FPT time
• APX is the class of problems with some constant-factor approximation algorithm
• Approximation-preserving reduction
• Exact algorithm