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 α away from the optimal solution, known as an α-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 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 f ( k ) n O ( 1 ) {\displaystyle f(k)n^{O(1)}} time, where f ( k ) {\displaystyle f(k)} is a function independent of the input size 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 α-approximation in f ( k ) n O ( 1 ) {\displaystyle f(k)n^{O(1)}} time, where f ( k ) {\displaystyle f(k)} is a function independent of the input size 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 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 is shown to admit an α-approximation algorithm running in f ( k ) n O ( 1 ) {\displaystyle f(k)n^{O(1)}} time, while in contrast the problem neither has a polynomial-time α-approximation algorithm (under some complexity assumption, e.g., P ≠ N P {\displaystyle {\mathsf {P}}\neq {\mathsf {NP}}} ), nor an FPT algorithm for the given parameter 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 ε > 0 {\displaystyle \varepsilon >0} a ( 1 + ε ) {\displaystyle (1+\varepsilon )} -approximation can be computed in f ( k , ε ) n g ( ε ) {\displaystyle f(k,\varepsilon )n^{g(\varepsilon )}} time for some functions f and 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 k. Since the degree of the polynomial in the runtime of a PAS depends on a function g ( ε ) {\displaystyle g(\varepsilon )} , the value of ε {\displaystyle \varepsilon } is assumed to be arbitrary but constant in order for the PAS to run in FPT time. If this assumption is unsatisfying, ε {\displaystyle \varepsilon } is treated as a parameter as well to obtain an efficient parameterized approximation scheme (EPAS), which for any ε > 0 {\displaystyle \varepsilon >0} computes a ( 1 + ε ) {\displaystyle (1+\varepsilon )} -approximation in f ( k , ε ) n O ( 1 ) {\displaystyle f(k,\varepsilon )n^{O(1)}} time for some function f. This is similar in spirit to an efficient polynomial-time approximation scheme (EPTAS).
k-Cut The k-cut problem has no polynomial-time ( 2 − ε ) {\displaystyle (2-\varepsilon )} -approximation algorithm for any ε > 0 {\displaystyle \varepsilon >0} , assuming P ≠ N P {\displaystyle {\mathsf {P}}\neq {\mathsf {NP}}} and the small set expansion hypothesis. It is also W[1]-hard parameterized by the number k of required components. However an EPAS exists, which computes a ( 1 + ε ) {\displaystyle (1+\varepsilon )} -approximation in ( k / ε ) O ( k ) n O ( 1 ) {\displaystyle (k/\varepsilon )^{O(k)}n^{O(1)}} time.
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