Abstract. It is known to be NP-hard to decide whether a graph can be made chordal by the deletion of k vertices. Here we present a uniformly polynomial-time algorithm for the problem: the running time is f(k) ·n α for some constant α not depending on k and some f depending only on k. For large values of n, such an algorithm is much better than trying all the O(n k) possibilities. Therefore, the chordal deletion problem parameterized by the number k of vertices to be deleted is fixed-parameter tractable. This answers an open question of Cai [2]. 1

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.

Research on parameterized algorithmics for NP-hard problems has steadily grown over the last years. We survey and discuss how parameterized complexity analysis naturally develops into the field of multivariate algorithmics. Correspondingly, we describe how to perform a systematic investigation and exploitation of the “parameter space” of computationally hard problems.

Inspired by a real-life application, we investigate the computationally hard problem of extending a precoloring of an interval graph to a proper coloring under some bound on the number of available colors. We are interested in quickly determining whether or not such an extension exists on instances occurring in practice in connection with campsite bookings on a campground. A naive exhaustive search does not terminate in reasonable time. We have formulated a new approach which moves the computation time within the usable range on all the data samples available to us.

Abstract. Incrementally k-list coloring a graph means that a graph is given by adding stepwise one vertex after another, and for each intermediate step we ask for a vertex coloring such that each vertex has one of the colors specified by its associated list containing some of in total k colors. We introduce the “conservative version ” of this problem by adding a further parameter c ∈ � specifying the maximum number of vertices to be recolored between two subsequent graphs (differing by one vertex). This “conservation parameter ” c models the natural quest for a modest evolution of the coloring in the course of the incremental process instead of performing radical changes. We show that the problem is NP-hard for k ≥ 3 and W[1]-hard when parameterized by c. In contrast, the problem becomes fixed-parameter tractable with respect to the combined parameter (k, c). We prove that the problem has an exponential-size kernel with respect to (k, c) and there is no polynomial-size kernel unless NP ⊆ coNP/poly. Finally, we provide empirical findings for the practical relevance of our approach in terms of a very effective graph coloring heuristic.

Abstract. Research on parameterized algorithmics for NP-hard problems has steadily grown over the last years. We survey and discuss how parameterized complexity analysis naturally develops into the field of multivariate algorithmics. Correspondingly, we describe how to perform a systematic investigation and exploitation of the “parameter space ” of computationally hard problems.