A parameterized approximation algorithm is a type of
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
that aims to find approximate solutions to
NP-hard
In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard pr ...
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
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 solut ...
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 and the more recent survey by Feldmann et al.
Obtainable approximation ratios
The full potential of parameterized approximation algorithms is utilized when a given
optimization problem
In mathematics, computer science and economics, an optimization problem is the problem of finding the ''best'' solution from all feasible solutions.
Optimization problems can be divided into two categories, depending on whether the variables ...
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 hard">hard).
For example, some problems that are
APX-hard and
W hard">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).
''k''-cut
The
''k''-cut problem has no polynomial-time
-approximation algorithm for any
, assuming
and the
small set expansion hypothesis
The small set expansion hypothesis or small set expansion conjecture in computational complexity theory is an unproven computational hardness assumption. Under the small set expansion hypothesis it is assumed to be computationally infeasible to d ...
. It is also W
hard parameterized by the number
of required components. However an EPAS exists, which computes a
-approximation in
time.
Steiner Tree
The
Steiner tree problem
In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of problems in combinatorial optimization. While Steiner tree problems may be formulated in a ...
is FPT parameterized by the number of terminals. However for the "dual" parameter consisting of the number
of non-terminals contained in the optimum solution, the problem is
W hard">hard (due to a folklore reduction from the
Dominating Set problem). Steiner Tree is also known to be
APX-hard. However, there is an EPAS computing a
-approximation in
time.
Strongly-connected Steiner subgraph
It is known that the Strongly Connected Steiner Subgraph problem is W
hard parameterized by the number
of terminals, and also does not admit an
-approximation in polynomial time (under standard
complexity assumptions). However a 2-approximation can be computed in
time. Furthermore, this is best possible, since no
-approximation can be computed in
time for any function
, under Gap-
ETH
(colloquially)
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, image = ETHZ.JPG
, image_size =
, established =
, type = Public
, budget = CHF 1.896 billion (2021)
, rector = Günther Dissertori
, president = Joël Mesot
, ac ...
.
''k''-median and ''k''-means
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
(colloquially)
, former_name = eidgenössische polytechnische Schule
, image = ETHZ.JPG
, image_size =
, established =
, type = Public
, budget = CHF 1.896 billion (2021)
, rector = Günther Dissertori
, president = Joël Mesot
, ac ...
.
Matching parameterized approximation algorithms exist,
[ 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
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
of the underlying metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
. In the Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean sp ...
, the k-Median and k-Means problems admit an EPAS parameterized by the dimension , and also an EPAS parameterized by . The former was generalized to an EPAS for the parameterization by the doubling dimension. For the loosely related highway dimension parameter, only an approximation scheme with XP runtime is known to date.
''k''-center
For the metric ''k''-center problem a 2-approximation can be computed in polynomial time. However, when parameterizing by either the number of centers, the doubling dimension (in fact the dimension of a Manhattan metric
A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian co ...
), or the highway dimension,[ no parameterized -approximation algorithm exists, under standard complexity assumptions. Furthermore, the k-Center problem is W hard even on ]planar graphs
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross ...
when simultaneously parameterizing it by the number of centers, the doubling dimension, the highway dimension, and the pathwidth
In graph theory, a path decomposition of a graph is, informally, a representation of as a "thickened" path graph, and the pathwidth of is a number that measures how much the path was thickened to form . More formally, a path-decomposition ...
. However, when combining with the doubling dimension an EPAS exists,[ and the same is true when combining with the highway dimension. 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. Regarding the pathwidth, k-Center admits an EPAS even for the more general ]treewidth
In graph theory, the treewidth of an undirected graph is an integer number which specifies, informally, how far the graph is from being a tree. The smallest treewidth is 1; the graphs with treewidth 1 are exactly the trees and the forests. The gr ...
parameter, and also for cliquewidth
In graph theory, the clique-width of a Graph (discrete mathematics), graph is a parameter that describes the structural complexity of the graph; it is closely related to treewidth, but unlike treewidth it can be bounded even for dense graphs.
It ...
.
Densest subgraph
An optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criterion, from some set of available alternatives. It is generally divided into two subfi ...
variant of the ''k''-Clique problem is the Densest ''k''-Subgraph problem (which is a 2-ary Constraint Satisfaction problem
Constraint satisfaction problems (CSPs) are mathematical questions defined as a set of objects whose state must satisfy a number of constraints or limitations. CSPs represent the entities in a problem as a homogeneous collection of finite constr ...
), 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
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related contexts, ...
optimal, since under Gap-ETH
(colloquially)
, former_name = eidgenössische polytechnische Schule
, image = ETHZ.JPG
, image_size =
, established =
, type = Public
, budget = CHF 1.896 billion (2021)
, rector = Günther Dissertori
, president = Joël Mesot
, ac ...
no -approximation can be computed in FPT time parameterized by .
Dominating set
For the Dominating set problem it is W hard to compute any -approximation in time for any functions and .
Approximate kernelization
Kernelization
In computer science, a kernelization is a technique for designing efficient algorithms that achieve their efficiency by a preprocessing stage in which inputs to the algorithm are replaced by a smaller input, called a "kernel". The result of solvi ...
is a technique used in fixed-parameter tractability
In computer science, parameterized complexity is a branch of computational complexity theory that focuses on classifying computational problems according to their inherent difficulty with respect to ''multiple'' parameters of the input or output. T ...
to pre-process an instance of an NP-hard
In computational complexity theory, NP-hardness ( non-deterministic polynomial-time hardness) is the defining property of a class of problems that are informally "at least as hard as the hardest problems in NP". A simple example of an NP-hard pr ...
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., but there are other related notions in the literature such as Turing kernels and -fidelity kernelization.
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.[ 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.][ 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.][ When parameterizing Steiner Tree by the number of non-terminals in the optimum solution, the problem is W hard (and thus admits no exact kernel at all, unless FPT=W , but still admits a PSAKS.][
]
References
{{reflist
Algorithms
Approximation algorithms
Parameterized complexity