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 ${\displaystyle \alpha }$ away from the optimal solution, known as an ${\displaystyle \alpha }$-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 ${\displaystyle k}$. 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 ${\displaystyle f(k)n^{O(1)}}$ time, where ${\displaystyle f(k)}$ is a function independent of the input size ${\displaystyle n}$.

A parameterized approximation algorithm aims to find a balance between these two approaches by finding approximate solutions in FPT time: the algorithm computes an ${\displaystyle \alpha }$-approximation in ${\displaystyle f(k)n^{O(1)}}$ time, where ${\displaystyle f(k)}$ is a function independent of the input size ${\displaystyle n}$. 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]

Obtainable approximation ratios

The full potential of parameterized approximation algorithms is utilized when a given optimization problem is shown to admit an ${\displaystyle \alpha }$-approximation algorithm running in ${\displaystyle f(k)n^{O(1)}}$ time, while in contrast the problem neither has a polynomial-time ${\displaystyle \alpha }$-approximation algorithm (under some complexity assumption, e.g., ${\displaystyle P\neq NP}$), nor an FPT algorithm for the given parameter ${\displaystyle k}$ (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 ${\displaystyle \varepsilon >0}$ a ${\displaystyle (1+\varepsilon )}$-approximation can be computed in ${\displaystyle f(k,\varepsilon )n^{g(\varepsilon )}}$ time for some functions ${\displaystyle f}$ and ${\displaystyle g}$. 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 ${\displaystyle k}$. Since the degree of the polynomial in the runtime of a PAS depends on a function ${\displaystyle g(\varepsilon )}$, the value of ${\displaystyle \varepsilon }$ is assumed to be arbitrary but constant in order for the PAS to run in FPT time. If this assumption is unsatisfying, ${\displaystyle \varepsilon }$ is treated as a parameter as well to obtain an efficient parameterized approximation scheme (EPAS), which for any ${\displaystyle \varepsilon >0}$ computes a ${\displaystyle (1+\varepsilon )}$-approximation in ${\displaystyle f(k,\varepsilon )n^{O(1)}}$ time for some function ${\displaystyle f}$. This is similar in spirit to an efficient polynomial-time approximation scheme (EPTAS).

Approximate kernelization

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 ${\displaystyle I}$ and a parameter ${\displaystyle k}$, and returns a new instance ${\displaystyle I'}$ with parameter ${\displaystyle k'}$ such that the size of ${\displaystyle I'}$ and ${\displaystyle k'}$ is bounded as a function of the input parameter ${\displaystyle k}$, and the algorithm runs in polynomial time. An ${\displaystyle \alpha }$-approximate kernelization algorithm is a variation of this technique that is used in parameterized approximation algorithms. It returns a kernel ${\displaystyle I'}$ such that any ${\displaystyle \beta }$-approximation in ${\displaystyle I'}$ can be converted into an ${\displaystyle \alpha \beta }$-approximation to the input instance ${\displaystyle I}$ 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 ${\displaystyle \alpha }$-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 ${\displaystyle \alpha }$-approximate kernelization algorithm that computes a polynomial-sized kernel and for which ${\displaystyle \alpha }$ can be set to ${\displaystyle 1+\varepsilon }$ for any ${\displaystyle \varepsilon >0}$.

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 ${\displaystyle NP\subseteq coNP/poly}$), 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 ${\displaystyle NP\subseteq coNP/poly}$), 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]

Talks on parameterized approximations

• Daniel Lokshtanov: A Parameterized Approximation Scheme for k-Min Cut
• Tuukka Korhonen: Single-Exponential Time 2-Approximation Algorithm for Treewidth
• Karthik C. S.: Recent Hardness of Approximation results in Parameterized Complexity
• Ariel Kulik. Two-variable Recurrence Relations with Application to Parameterized Approximations
• Meirav Zehavi. FPT Approximation
• Vincent Cohen-Added: On the Parameterized Complexity of Various Clustering Problems
• Fahad Panolan. Parameterized Approximation for Independent Set of Rectangles
• Andreas Emil Feldmann. Approximate Kernelization Schemes for Steiner Networks

References