We consider parameterized problems where some separation property has to be achieved by deleting as few vertices as possible. The following five problems are studied: delete k vertices such that (a) each of the given ℓ terminals is separated from the others, (b) each of the given ℓ pairs of terminals is separated, (c) exactly ℓ vertices are cut away from the graph, (d) exactly ℓ connected vertices are cut away from the graph, (e) the graph is separated into at least ℓ components. We show that if both k and ℓ are

We develop new techniques for deriving strong computational lower bounds for a class of well-known NP-hard problems. This class includes weighted satisfiability, dominating set, hitting set, set cover, clique, and independent set. For example, although a trivial enumeration can easily test in time O(n k) if a given graph of n vertices has a clique of size k, we prove that unless an unlikely collapse occurs in parameterized complexity theory, the problem is not solvable in time f(k)n o(k) for any function f, even if we restrict the parameter values to be bounded by an arbitrarily small function of n. Under the same assumption, we prove that even if we restrict the parameter values k to be of the order Θ(µ(n)) for any reasonable function µ, no algorithm of running time n o(k) can test if a graph of n vertices has a clique of size k. Similar strong lower bounds on the computational complexity are also derived for other NP-hard problems in the above class. Our techniques can be further extended to derive computational lower bounds on polynomial time approximation schemes for NP-hard optimization problems. For example, we prove that the NP-hard distinguishing substring selection problem, for which a polynomial time approximation scheme has been recently developed, has no polynomial time approximation schemes of running time f(1/ɛ)n o(1/ɛ) for any function f unless an unlikely collapse occurs in parameterized complexity theory.

In the precoloring extension problem (PrExt) a graph is given with some of the vertices having preassigned colors and it has to be decided whether this coloring can be extended to a proper coloring of the graph with the given number of colors. Two parameterized versions of the problem are studied in the paper: either the number of precolored vertices or the number of colors used in the precoloring is restricted to be at most k. We show that for chordal graphs these problems are polynomial-time solvable for every fixed k, but W[1]-hard if k is the parameter. For a graph class F, let F + ke (resp., F +kv) denote those graphs that can be made to be a member of F by deleting at most k edges (resp., vertices). We investigate the connection between PrExt in F (with the two parameters defined above) and the coloring of F + ke, F + kv graphs (with k being the parameter). Answering an open question of Leizhen Cai [5], we show that coloring chordal+ke graphs is fixed-parameter tractable. 1

We show that searching the k-change neighborhood is W[1]-hard for metric TSP, which means that finding the best tour in the k-change neighborhood essentially requires complete search (modulo some complexitytheoretic assumptions). Keywords: Traveling Salesperson Problem, W[1]-hardness, parameterized complexity, local search

This paper surveys parameterized complexity results for hard geometric algorithmic problems. It includes fixed-parameter tractable problems in graph drawing, geometric graphs, geometric covering and several other areas, together with an overview of the algorithmic techniques used. Fixed-parameter intractability results are surveyed as well. Finally, we give some directions for future research.

Abstract. In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest. A well-developed example is the investigation of how the parameter treewidth influences the complexity of (other) graph problems. The reason why such investigations are of general interest is that real-world input distributions for computational problems often inherit structure from the natural computational processes that produce the problem instances (not necessarily in obvious, or well-understood ways). The max leaf number ml(G) of a connected graph G is the maximum number of leaves in a spanning tree for G. Exploring questions analogous to the well-studied case of treewidth, we can ask: how hard is it to solve 3-Coloring, Hamilton Path, Minimum Dominating Set, Minimum Bandwidth or many other problems, for graphs of bounded max leaf number? What optimization problems are W [1]-hard under this parameterization? We do two things: (1) We describe much improved FPT algorithms for a large number of graph problems, for input graphs G for which ml(G) ≤ k, based on the polynomial-time extremal structure theory canonically associated to this parameter. We consider improved algorithms both from the point of view of kernelization bounds, and in terms of improved fixed-parameter tractable (FPT) runtimes O ∗ (f(k)). (2) The way that we obtain these concrete algorithmic results is general and systematic. We describe the approach, and raise programmatic questions. 1

Abstract. We establish a close connection between (sub)exponential time complexity and parameterized complexity by proving that the so-called miniaturization mapping is a reduction preserving isomorphism between the two theories. Key words. parameterized complexity, exponential time complexity, exponential time hypothesis, subexponential time

Abstract. 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. Key words. Fixed-parameter tractability, approximation algorithms, hardness of approximation. 1

Abstract We present an algorithm for the 3-center problem in (Rd, L1), i. e., for finding the smallest side length for 3 cubes that cover a given n-point set in Rd, that runs in O(n log n)time for any fixed dimension d. This shows that the problem is fixed-parameter tractable when parameterized with d. Onthe other hand, using tools from parameterized complexity theory, we show that this is unlikely to be the case with the k-center problem in (Rd, L2), for any k> = 2. In particular, we prove that deciding whether a given n-point set in Rdcan be covered by the union of 2 balls of given radius is W[1]-hard with respect to d, and thus not fixed-parametertractable unless FPT=W[1]. Our reduction also shows that even an O(no(d))-time algorithm for the latter does not exist, unless SNP ae DTIME(2o(n)).

We construct a new public key encryption based on two assumptions: 1. One can obtain a pseudorandom generator with small locality by connecting the outputs to the inputs using any sufficiently good unbalanced expander. 2. It is hard to distinguish between a random graph that is such an expander and a random graph where a (planted) random logarithmic-sized subset S of the outputs is connected to fewer than |S | inputs. The validity and strength of the assumptions raise interesting new algorithmic and pseudorandomness questions, and we explore their relation to the current state-of-art. 1