A **parameterized approximation algorithm** is a type of
algorithm that aims to find approximate solutions to
NP-hard
optimization problems in polynomial time in the input size and a function of a specific parameter. These algorithms are designed to combine the best aspects of both traditional
approximation algorithms and
fixed-parameter tractability.

In traditional approximation algorithms, the goal is to find solutions that are at most a certain factor away from the optimal solution, known as an -approximation, in polynomial time. On the other hand, parameterized algorithms are designed to find exact solutions to problems, but with the constraint that the running time of the algorithm is polynomial in the input size and a function of a specific parameter . The parameter describes some property of the input and is small in typical applications. The problem is said to be fixed-parameter tractable (FPT) if there is an algorithm that can find the optimum solution in time, where is a function independent of the input size .

A parameterized approximation algorithm aims to find a balance between these two approaches by finding approximate solutions in FPT time: the algorithm computes an -approximation in time, where is a function independent of the input size . This approach aims to overcome the limitations of both traditional approaches by having stronger guarantees on the solution quality compared to traditional approximations while still having efficient running times as in FPT algorithms. An overview of the research area studying parameterized approximation algorithms can be found in the survey of Marx^{
[1]} and the more recent survey by Feldmann et al.^{
[2]}

The full potential of parameterized approximation algorithms is utilized when a given optimization problem is shown to admit an -approximation algorithm running in time, while in contrast the problem neither has a polynomial-time -approximation algorithm (under some complexity assumption, e.g., ), nor an FPT algorithm for the given parameter (i.e., it is at least W[1]-hard).

For example, some problems that are
APX-hard and
W[1]-hard admit a **parameterized approximation scheme (PAS)**, i.e., for any a -approximation can be computed in time for some functions and . This then circumvents the lower bounds in terms of polynomial-time approximation and fixed-parameter tractability. A PAS is similar in spirit to a
polynomial-time approximation scheme (PTAS) but additionally exploits a given parameter . Since the degree of the polynomial in the runtime of a PAS depends on a function , the value of is assumed to be arbitrary but constant in order for the PAS to run in FPT time. If this assumption is unsatisfying, is treated as a parameter as well to obtain an **efficient** **parameterized approximation scheme (EPAS)**, which for any computes a -approximation in time for some function . This is similar in spirit to an
efficient polynomial-time approximation scheme (EPTAS).

The
k-cut problem has no polynomial-time -approximation algorithm for any , assuming and the
small set expansion hypothesis.^{
[3]} It is also W[1]-hard parameterized by the number of required components.^{
[4]} However an EPAS exists, which computes a -approximation in time.^{
[5]}

The
Steiner Tree problem is FPT parameterized by the number of terminals.^{
[6]} However for the "dual" parameter consisting of the number of non-terminals contained in the optimum solution, the problem is
W[2]-hard (due to a folklore reduction from the
Dominating Set problem). Steiner Tree is also known to be
APX-hard.^{
[7]} However, there is an EPAS computing a -approximation in time.^{
[8]}

It is known that the Strongly Connected Steiner Subgraph problem is W[1]-hard parameterized by the number of terminals,^{
[9]} and also does not admit an -approximation in polynomial time (under standard
complexity assumptions).^{
[10]} However a 2-approximation can be computed in time.^{
[11]} Furthermore, this is best possible, since no -approximation can be computed in time for any function , under Gap-
ETH.^{
[12]}

For the well-studied metric clustering problems of
k-Median and
k-Means parameterized by the number of centers, it is known that no -approximation for k-Median and no -approximation for k-Means can be computed in time for any function , under Gap-
ETH.^{
[13]} Matching parameterized approximation algorithms exist,^{
[13]} but it is not known whether matching approximations can be computed in polynomial time.

Clustering is often considered in settings of low dimensional data, and thus a practically relevant parameterization is by the
dimension of the underlying
metric. In the
Euclidean space, the k-Median and k-Means problems admit an EPAS parameterized by the dimension ,^{
[14]}^{
[15]} and also an EPAS parameterized by .^{
[16]}^{
[17]} The former was generalized to an EPAS for the parameterization by the
doubling dimension.^{
[18]} For the loosely related highway dimension parameter, only an approximation scheme with
XP runtime is known to date.^{
[19]}

For the metric
k-Center problem a 2-approximation can be computed in polynomial time. However, when parameterizing by either the number of centers,^{
[20]} the
doubling dimension (in fact the dimension of a
Manhattan metric),^{
[21]} or the highway dimension,^{
[20]} no parameterized -approximation algorithm exists, under standard
complexity assumptions. Furthermore, the k-Center problem is W[1]-hard even on
planar graphs when simultaneously parameterizing it by the number of centers, the
doubling dimension, the highway dimension, and the
pathwidth.^{
[22]} However, when combining with the doubling dimension an EPAS exists,^{
[22]} and the same is true when combining with the highway dimension.^{
[23]} For the more general version with vertex capacities, an EPAS exists for the parameterization by k and the doubling dimension, but not when using k and the highway dimension as the parameter.^{
[24]} Regarding the pathwidth, k-Center admits an EPAS even for the more general
treewidth parameter, and also for
cliquewidth.^{
[25]}

An
optimization variant of the
k-Clique problem is the Densest k-Subgraph problem (which is a 2-ary
Constraint Satisfaction problem), where the task is to find a subgraph on vertices with maximum number of edges. It is not hard to obtain a -approximation by just picking a
matching of size in the given input graph, since the maximum number of edges on vertices is always at most . This is also
asymptotically optimal, since under Gap-
ETH no -approximation can be computed in FPT time parameterized by .^{
[26]}

For the
Dominating Set problem it is W[1]-hard to compute any -approximation in time for any functions and .^{
[27]}

Kernelization is a technique used in
fixed-parameter tractability to pre-process an instance of an
NP-hard problem in order to remove "easy parts" and reveal the NP-hard core of the instance. A kernelization algorithm takes an instance and a parameter , and returns a new instance with parameter such that the size of and is bounded as a function of the input parameter , and the algorithm runs in polynomial time. An **-approximate kernelization algorithm** is a variation of this technique that is used in parameterized approximation algorithms. It returns a kernel such that any -approximation in can be converted into an -approximation to the input instance in polynomial time. This notion was introduced by Lokshtanov et al.,^{
[28]} but there are other related notions in the literature such as Turing kernels^{
[29]} and **-fidelity kernelization.**^{
[30]}

As for regular (non-approximate) kernels, a problem admits an α-approximate kernelization algorithm if and only if it has a parameterized α-approximation algorithm. The proof of this fact is very similar to
the one for regular kernels.^{
[28]} However the guaranteed approximate kernel might be of exponential size (or worse) in the input parameter. Hence it becomes interesting to find problems that admit polynomial sized approximate kernels. Furthermore, a **polynomial-sized approximate kernelization scheme (PSAKS)** is an **-approximate kernelization algorithm that computes a polynomial-sized kernel and for which **** can be set to for any .
**

For example, while the Connected Vertex Cover problem is FPT parameterized by the solution size, it does not admit a (regular) polynomial sized kernel (unless ), but a PSAKS exists.^{
[28]} Similarly, the Steiner Tree problem is FPT parameterized by the number of terminals, does not admit a polynomial sized kernel (unless ), but a PSAKS exists.^{
[28]} When parameterizing Steiner Tree by the number of non-terminals in the optimum solution, the problem is W[2]-hard (and thus admits no exact kernel at all, unless FPT=W[2]), but still admits a PSAKS.^{
[8]}

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