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.
For the special case of where all input objects are squares a PTAS is known @cite_16 but there can be no EPTAS @cite_27 . Recently, @cite_1 found polynomial-time algorithms for and with approximation ratio smaller than @math (also for the weighted case). For the special case that all input objects are squares there is a PTAS @cite_19 and even an EPTAS @cite_15 .
@cite_1 @cite_19 @cite_27 @cite_15 @cite_16
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An EPTAS (efficient PTAS) is an approximation scheme where e does not appear in the exponent of n, i.e., the running time is f(e)nc. We use parameterized complexity to investigate the possibility of improving the known approximation schemes for certain geometric problems to EPTAS. Answering an open question of Alber and Fiala [2], we show that Maximum Independent Set is W[1]-complete for the intersection graphs of unit disks and axis-parallel unit squares in the plane. A standard consequence of this result is that the @math time PTAS of [11] for Maximum Independent Set on unit disk graphs cannot be improved to an EPTAS. Similar results are obtained for the problem of covering points with squares. A polynomial time approximation scheme (PTAS) for an optimization problem A is an algorithm that given in input an instance of A and e > 0 finds a (1 + e)-approximate solution in time that is polynomial for each fixed e. Typical running times are nO(1e) or 21eO(1) n. While algorithms of the former kind tend to be impractical, the latter ones are more interesting. In several cases, the development of algorithms of the second type required considerably new, and sometimes harder, techniques. For some interesting problems, only nO(1e) approximation schemes are known. Under likely assumptions, we prove that for some problems (including natural ones) there cannot be approximation schemes running in time f(1 ϵ) n p0(1), no matter how fast function f grows. Our result relies on a connection with Parameterized Complexity Theory, and we show that this connection is necessary. An important question in theoretical computer science is to determine the best possible running time for solving a problem at hand. For geometric optimization problems, we often understand their complexity on a rough scale, but not very well on a finer scale. One such example is the two-dimensional knapsack problem for squares. There is a polynomial time (1 + ϵ)-approximation algorithm for it (i.e., a PTAS) but the running time of this algorithm is triple exponential in 1 ϵ, i.e., Ω(n221 ϵ). A double or triple exponential dependence on 1 ϵ is inherent in how this and several other algorithms for other geometric problems work. In this paper, we present an EPTAS for knapsack for squares, i.e., a (1+ϵ)-approximation algorithm with a running time of Oϵ(1)·nO(1). In particular, the exponent of n in the running time does not depend on ϵ at all! Since there can be no FPTAS for the problem (unless P = NP) this is the best kind of approximation scheme we can hope for. To achieve this improvement, we introduce two new key ideas: We present a fast method to guess the Ω(221 ϵ) relatively large squares of a suitable near-optimal packing instead of using brute-force enumeration. Secondly, we introduce an indirect guessing framework to define sizes of cells for the remaining squares. In the previous PTAS each of these steps needs a running time of Ω(n221 ϵ) and we improve both to Oϵ(1) · nO(1). We complete our result by giving an algorithm for two-dimensional knapsack for rectangles under (1 + ϵ)-resource augmentation. In this setting, we also improve the best known running time of Ω(n1 ϵ1 ϵ) to Oϵ(1) · nO(1) and compute even a solution with optimal profit, in contrast to the best previously known polynomial time algorithm for this setting that computes only an approximation. We believe that our new techniques have the potential to be useful for other settings as well. In the Closest String problem one is given a family S of equal-length strings over some fixed alphabet, and the task is to find a string y that minimizes the maximum Hamming distance between y and a string from S. While polynomial-time approximation schemes (PTASes) for this problem are known for a long time [; J. ACM'02], no efficient polynomial-time approximation scheme (EPTAS) has been proposed so far. In this paper, we prove that the existence of an EPTAS for Closest String is in fact unlikely, as it would imply that FPT=W[1], a highly unexpected collapse in the hierarchy of parameterized complexity classes. Our proof also shows that the existence of a PTAS for Closest String with running time f(eps) n^o(1 eps), for any computable function f, would contradict the Exponential Time Hypothesis.