A Fixed-Parameter Tractable (FPT) ρ-approximation algorithm for a minimization (resp. maximization) parameterized problem P is an FPT-algorithm that, given an instance (x, k) ∈ P computes a solution of cost at most k · ρ(k) (resp. k/ρ(k)) if a solution of cost at most (resp. at least) k exists; otherwise the output can be arbitrary. For well-known intractable problems such as the W[1]-hard Cliq...

There is a computable function f such that the problem of deciding, given a word w and an MSO sentence φ whether, S w |= φ can be decided in time O(f (l)n) where l is the length of φ and n is the length of w.

As a generalization of paths, the notion of paths of bandwidth w is introduced. We show that, for constant w ≥ 1, the corresponding search problem for such a path of length k in a given graph is NP-complete and fixed-parameter tractable (FPT) in the parameter k, like this is known for the special case w = 1, the LONGEST PATH problem. We state the FPT algorithm in terms of a guess and check prot...

We show that for any fixed i, j ≥ 1, the k-Dominating Set problem restricted to graphs that do not have Ki,j as a subgraph is fixed parameter tractable (FPT) and has a polynomial kernel. This result implies that this problem restricted to bounded-degenerate graphs has a polynomial kernel, solving an open problem posed by Alon and Gutner in [3]. Our result extends the class of graphs for which t...

Given a set of obstacles and two points in the plane, is there a path between the two points that does not cross more than k different obstacles? Equivalently, can we remove k obstacles so that there is an obstacle-free path between the two designated points? This is a fundamental problem that has undergone a tremendous amount of work by researchers in various areas, including computational geo...

A knockout (or single-elimination) tournament is a format of competition that very popular in practice (particularly sports, elections and decision making), which has been extensively intensively studied from theoretical point view for more than decade. Particular attention devoted to the Tournament Fixing problem, where, roughly speaking, objective determine whether we can conduct way makes ou...