text: The area of parameterized approximation seeks to combine approximation and parameterized algorithms to obtain, e.g., (1+eps)-approximations in f(k,eps)n^ O(1) time where k is some parameter of the input. We obtain the following results on parameterized approximability: 1) In the maximum independent set of rectangles problem (MISR) we are given a collection of n axis parallel rectangles in the plane. Our goal is to select a maximum-cardinality subset of pairwise non-overlapping rectangles. This problem is NP-hard and also W[1]-hard [Marx, ESA'05]. The best-known polynomial-time approximation factor is O(loglog n) [Chalermsook and Chuzhoy, SODA'09] and it admits a QPTAS [Adamaszek and Wiese, FOCS'13; Chuzhoy and Ene, FOCS'16]. Here we present a parameterized approximation scheme (PAS) for MISR, i.e. an algorithm that, for any given constant eps>0 and integer k>0, in time f(k,eps)n^ g(eps) , either outputs a solution of size at least k (1+eps), or declares that the optimum solution has size less than k. 2) In the (2-dimensional) geometric knapsack problem (TDK) we are given an axis-aligned square knapsack and a collection of axis-aligned rectangles in the plane (items). Our goal is to translate a maximum cardinality subset of items into the knapsack so that the selected items do not overlap. In the version of TDK with rotations (TDKR), we are allowed to rotate items by 90 degrees. Both variants are NP-hard, and the best-known polynomial-time approximation factors are 558 325+eps and 4 3+eps, resp. [, FOCS'17]. These problems admit a QPTAS for polynomially bounded item sizes [Adamaszek and Wiese, SODA'15]. We show that both variants are W[1]-hard. Furthermore, we present a PAS for TDKR. For all considered problems, getting time f(k,eps)n^ O(1) , rather than f(k,eps)n^ g(eps) , would give FPT time f'(k)n^ O(1) exact algorithms using eps=1 (k+1), contradicting W[1]-hardness.
  @cite_24: Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We discuss the different ways parameterized complexity can be extended to approximation algorithms, survey results of this type and propose directions for future research.
  @cite_5: Combining classical approximability questions with parameterized complexity, we introduce a theory of parameterized approximability. The main intention of this theory is to deal with the efficient approximation of small cost solutions for optimisation problems.
  @cite_10: The notion of fixed-parameter approximation is introduced to investigate the approximability of optimization problems within the framework of fixed-parameter computation. This work partially aims at enhancing the world of fixed-parameter computation in parallel with the conventional theory of computation that includes both exact and approximate computations. In particular, it is proved that fixed-parameter approximability is closely related to the approximation of small-cost solutions in polynomial time. It is also demonstrated that many fixed-parameter intractable problems are not fixed-parameter approximable. On the other hand, fixed-parameter approximation appears to be a viable approach to solving some inapproximable yet important optimization problems. For instance, all problems in the class MAX SNP admit fixed-parameter approximation schemes in time O(2 @math p(n)) for any small e> 0.
  @cite_7: Up to now, most work in the area of parameterized complexity has focussed on exact algorithms for decision problems. The goal of this paper is to apply parameterized ideas to approximation. We begin exploration of parameterized approximation problems, where the problem in question is a parameterized decision problem, and the required approximation factor is treated as a second parameter for the problem.
summary: The systematic study of parameterized approximation as a field was initiated independently by three separate publications @cite_10 @cite_5 @cite_7 . A very good introduction to the area including key definitions as well as a survey of earlier results that fit into the picture was given by Marx @cite_24 . In particular, Marx also defined a so-called that, given input @math will run for @math time and return (say, for a maximization problem) a solution of value at least @math if the optimum is at least @math . As mentioned earlier, Marx pointed out that a standard FPT-approximation scheme that finds a solution of value at least @math in time @math if @math is not interesting to study: By setting @math we can decide the decision problem @math ?'' in FPT time. Thus, such a scheme is not helpful if the decision problem is 1 -hard and therefore unlikely to have an FPT-algorithm. Nevertheless, PASs can be useful in this case, as they imply standard FPT-approximation algorithms with ratio @math for each fixed @math despite 1 -hardness.