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. 1.

We show for several planar and geometric problems that the best known approximation schemes are essentially optimal with respect to the dependence on ǫ. For example, we show that the 2O(1/ǫ) · n time approximation schemes for planar MAXIMUM INDEPENDENT SET and for TSP on a metric defined by a planar graph are essentially optimal: if there is a δ> 0 such that any of these problems admits a 2O((1/ǫ)1−δ) O(1) n time PTAS, then the Exponential Time Hypothesis (ETH) fails. It is known that MAXIMUM IN-DEPENDENT SET on unit disk graphs and the planar logic problems MPSAT, TMIN, TMAX admit nO(1/ǫ) time approximation schemes. We show that they are optimal in the sense that if there is a δ> 0 such that any of these problems admits a 2 (1/ǫ)O(1) nO((1/ǫ)1−δ) time PTAS, then ETH fails.

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Given a vertex-weighted directed graph G = (V,E) and a set T = {t1, t2,... tk} of k terminals, the objective of the STRONGLY CONNECTED STEINER SUBGRAPH (SCSS) problem is to find a vertex set H ⊆V of minimum weight such that G[H] contains a ti → t j path for each i 6 = j. The problem is NP-hard, but Feldman and Ruhl (FOCS ’99; SICOMP ’06) gave a novel nO(k) algorithm for the SCSS problem, where n is the number of vertices in the graph and k is the number of terminals. We explore how much easier the problem becomes on planar directed graphs. • Our main algorithmic result is a 2O(k logk) ·nO( k) al-gorithm for planar SCSS, which is an improvement of a factor of O( k) in the exponent over the algo-rithm of Feldman and Ruhl. • Our main hardness result is a matching lower bound for our algorithm: we show that planar SCSS does not have an f (k) · no( k) algorithm for any com-putable function f, unless the Exponential Time Hy-pothesis (ETH) fails. The algorithm eventually relies on the excluded grid theorem for planar graphs, but we stress that it is not simply a straightforward application of treewidth-based techniques: we need several layers of abstraction to arrive to a problem formulation where the speedup due to planarity can be exploited. To obtain the lower bound matching the algorithm, we need a delicate construction of gadgets arranged in a grid-like fashion to tightly control the number of terminals in the created instance.

Most interesting optimization problems on graphs are NP-hard, implying that (un-less P = NP) there is no polynomial time algorithm that solves all the instances of an NP-hard problem exactly. However, classical complexity measures the running time as a function of only the overall input size. The paradigm of parameterized complexity was introduced by Downey and Fellows to allow for a more refined multivariate analy-sis of the running time. In parameterized complexity, each problem comes along with a secondary measure k which is called the parameter. The goal of parameterized com-plexity is to design efficient algorithms for NP-hard problems when the parameter k is small, even if the input size is large. Formally, we say that a parameterized problem is fixed-parameter tractable (FPT) if instances of size n and parameter k can be solved in f (k) · nO(1) time, where f is a computable function which does not depend on n. A pa-rameterized problem belongs to the class XP if instances of size n and parameter k can be solved in f (k) ·nO(g(k)) time, where f and g are both computable functions. In this thesis we focus on the parameterized complexity of transversal and connec-tivity problems on directed graphs. This research direction has been hitherto relatively

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