The k-vertex disjoint paths problem is one of the most studied problems in algorithmic graph theory. In 1994, Schrijver proved that the problem can be solved in polynomial time for every fixed k when restricted to the class of planar digraphs and it was a long standing open question whether it is fixed-parameter tractable (with respect to parameter k) on this restricted class. Only recently, [1...

For many fixed-parameter problems that are trivially solvable in polynomial-time, such as k-Dominating Set, essentially no better algorithm is presently known than the one which tries all possible solutions. Other problems, such as k-Feedback Vertex Set, exhibit fixed-parameter tractability: for each fixed k the problem is solvable in time bounded by a polynomial of degree c, where c is a const...

We study the propagation of constraints that apply to a whole matrix of decision variables. We identify several cases where propagation is fixed parameter tractable, as well as other cases where propagation is intractable. We find that the number of rows (or columns) in the matrix as a useful parameter in describing the complexity of propagation of such matrix constraints.

For my final project I study the (nonparametric) unit demand pricing problem from a parametrized perspective. For instances with n items and m customers we give O(m) and O(n) algorithms respectively. We then investigate a price restricted variant of UDP called UDP-PS, and show the problem is fixed parameter tractable when the parameter is the number of customers.

A parameterized problem Π can be considered as a set of pairs (I, k) where I is the main part and k (usually an integer) is the parameter. Π is called fixed-parameter tractable (FPT) if membership of (I, k) in Π can be decided in time O(f(k)|I|), where |I| denotes the size of I , f(k) is a computable function, and c is a constant independent of k and I . An algorithm of complexity O(f(k)|I|c) i...

In this paper we study the parameterized complexity of two well-known permutation group problems which are NP-complete. • Given a permutation group G = 〈S〉 ≤ Sn and a parameter k, find a permutation π ∈ G such that |{i ∈ [n] | π(i) 6= i}| ≥ k. This generalizes the NP-complete problem of finding a fixed-point free permutation in G [CW10, Lub81] (this is the case when k = n). We show that this pr...

Disjunctive Answer Set Programming (ASP) is a powerful declarative programming paradigm whose main decision problems are located on the second level of the polynomial hierarchy. Identifying tractable fragments and developing efficient algorithms for such fragments are thus important objectives in order to complement the sophisticated ASP systems available to date. Hard problems can become tract...

ly speaking, what have we done in the previous section? After applying a number of rules in polynomial time to an instance of VERTEX COVER, we arrived at a reduced instance whose size can solely be expressed in terms of the parameter k. Since this can be easily done in O(n) time, we have found a data reduction for VERTEX COVER with guarantees concerning its running time as well as its effective...

The k-Coloring problem is to decide whether a graph can be colored with at most k colors such that no two adjacent vertices receive the same color. The List k-Coloring problem requires in addition that every vertex u must receive a color from some given set L(u) ⊆ {1, . . . , k}. Let Pn denote the path on n vertices, and G + H and rH the disjoint union of two graphs G and H and r copies of H, r...

We study an extension of first-order logic that allows to express cardinality conditions in a similar way as SQL’s COUNT operator. The corresponding logic FOC(P) was introduced by Kuske and Schweikardt [16], who showed that query evaluation for this logic is fixedparameter tractable on classes of databases of bounded degree. In the present paper, we first show that the fixed-parameter tractabil...