To solve NP-hard problems, polynomial-time preprocessing is a natural and promising approach. Preprocessing is based on data reduction techniques that take a problem's input instance and try to perform a reduction to a smaller, equivalent problem kernel. Problem kernelization is a methodology that is rooted in parameterized computational complexity. In this brief survey, we present data reduction and problem kernelization as a promising research field for algorithm and complexity theory.

We show that the NP-complete FEEDBACK VERTEX SET problem, which asks for the smallest set of vertices to remove from a graph to destroy all cycles, is deterministically solvable in O(c k · m) time. Here, m denotes the number of graph edges, k denotes the size of the feedback vertex set searched for, and c is a constant. We extend this to an algorithm enumerating all solutions in O(d k · m) time for a (larger) constant d. As a further result, we present a fixed-parameter algorithm with runtime O(2 k · m 2 ) for the NP-complete EDGE BIPARTIZATION problem, which asks for at most k edges to remove from a graph to make it bipartite.

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To cover the edges of a graph with a minimum number of cliques is an NP-hard problem with many applications. We develop for this problem efficient and effective polynomial-time data reduction rules that, combined with a search tree algorithm, allow for exact problem solutions in competitive time. This is confirmed by experiments with real-world and synthetic data. Moreover, we prove the fixed-parameter tractability of covering edges by cliques.

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We present exact algorithms for the NP-complete LONGEST COMMON SUBSEQUENCE problem for sequences with nested arc annotations, a problem occurring in structure comparison of RNA. Given two sequences of length at most n and nested arc structure, one of our algorithms determines (if existent) in O(3:31 k 1 +k 2 · n) time an arc-preserving subsequence of both sequences, which can be obtained by deleting (together with corresponding arcs) k1 letters from the ÿrst and k2 letters from the second sequence. A second algorithm shows that (in case of a four letter alphabet) we can ÿnd a length l arc-annotated subsequence in O(12 l · l · n) time. This means that the problem is ÿxed-parameter tractable when parameterized by the number of deletions as well as when parameterized by the subsequence length. Our ÿndings complement known approximation results which give a quadratic time factor-2-approximation for the general and polynomial A preliminary version of this paper titled "Towards optimally solving the LONGEST COMMON SUBSEQUENCE problem for sequences with nested arc annotations in linear time" was presented at the

The computation of Kemeny rankings is central to many applications in the context of rank aggregation. Given a set of permutations (votes) over a set of candidates, one searches for a "consensus permutation" that is "closest" to the given set of permutations. Unfortunately, the problem is NP-hard. We provide a broad study of the parameterized complexity for computing optimal Kemeny rankings. Beside the three obvious parameters "number of votes", "number of candidates", and solution size (called Kemeny score), we consider further structural parameterizations. More specifically, we show that the Kemeny score (and a corresponding Kemeny ranking) of an election can be computed efficiently whenever the average pairwise distance between two input votes is not too large. In other words, Kemeny Score is fixedparameter tractable with respect to the parameter "average pairwise Kendall-Tau distance d a ". We describe a fixed-parameter algorithm with running time 16 ⌈da⌉ • poly. Moreover, we extend our studies to the parameters "maximum range" and "average range" of positions a candidate takes in the input votes. Whereas Kemeny Score remains fixed-parameter tractable with respect to the parameter "maximum range", it becomes NP-complete in case of an average range of two. This excludes fixed-parameter tractability with respect to the parameter "average range" unless P=NP. Finally, we extend some of our results to votes with ties and incomplete votes, where in both cases one no longer has permutations as input.

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