We initiate the first systematic study of the NP-hard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixed-parameter algorithmics. In the unweighted case, one searches for a minimum number of vertex deletions to transform a graph into a collection of disjoint cliques. The parameter is the number of vertex deletions. We present efficient fixed-parameter algorithms for CVD applying the fairly new iterative compression technique. Moreover, we study the variant of CVD where the maximum number of cliques to be generated is prespecified. Here, we exploit connections to fixed-parameter algorithms for (weighted) Vertex Cover.

We consider ranking and clustering problems related to the aggregation of inconsistent information, in particular, rank aggregation, (weighted) feedback arc set in tournaments, consensus and correlation clustering, and hierarchical clustering. Ailon, Charikar, and Newman [4], Ailon and Charikar [3], and Ailon [2] proposed randomized constant factor approximation algorithms for these problems, which recursively generate a solution by choosing a random vertex as “pivot ” and dividing the remaining vertices into two groups based on the pivot vertex. In this paper, we answer an open question in these works by giving deterministic approximation algorithms for these problems. The analysis of our algorithms is simpler than the analysis of the randomized algorithms in [4], [3] and [2]. In addition, we consider the problem of finding minimum-cost rankings and clusterings which must obey certain constraints (e.g. an input partial order in the case of ranking problems), which were introduced by Hegde and Jain [25] (see also [34]). We show that the first type of algorithms we propose can also handle these constrained problems. In addition, we show that in the case of a rank aggregation or consensus clustering problem, if the input rankings or clusterings obey the constraints, then we can always ensure that the output of

Abstract. The Kemeny Score problem 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. Computing an optimal consensus permutation is NP-hard. We provide first, encouraging fixed-parameter tractability results for computing optimal scores (that is, the overall distance of an optimal consensus permutation). Our fixedparameter algorithms employ the parameters “score of the consensus”, “maximum distance between two input permutations”, and “number of candidates”. We extend our results to votes with ties and incomplete votes, thus, in both cases having no longer permutations as input. 1

Abstract. We introduce the s-Plex Editing problem generalizing the well-studied Cluster Editing problem, both being NP-hard and both being motivated by graph-based data clustering. Instead of transforming a given graph by a minimum number of edge modifications into a disjoint union of cliques (Cluster Editing), the task in the case of s-Plex Editing is now to transform a graph into a disjoint union of so-called s-plexes. Herein, an s-plex denotes a vertex set inducing a (sub)graph where every vertex has edges to all but at most s vertices in the s-plex. Cliques are 1-plexes. The advantage of s-plexes for s ≥ 2 is that they allow to model a more relaxed cluster notion (s-plexes instead of cliques), which better reflects inaccuracies of the input data. We develop a provably efficient and effective preprocessing based on data reduction (yielding a so-called problem kernel), a forbidden subgraph characterization of s-plex cluster graphs, and a depth-bounded search tree which is used to find optimal edge modification sets. Altogether, this yields efficient algorithms in case of moderate numbers of edge modifications. 1

This paper describes an efficient reduction of the learning problem of ranking to binary classification. The reduction is randomized and guarantees a pairwise misranking regret bounded by that of the binary classifier, improving on a recent result of Balcan et al. (2007) which ensures only twice that upper-bound. Moreover, our reduction applies to a broader class of ranking loss functions, admits a simple proof, and the expected time complexity of our algorithm in terms of number of calls to a classifier or preference function is also improved from Ω(n 2) to O(n log n). In addition, when the top k ranked elements only are required (k ≪ n), as in many applications in information extraction or search engine design, the time complexity of our algorithm can be further reduced to O(k log k+n). Our reduction and algorithm are thus practical for realistic applications where the number of points to rank exceeds several thousands. Much of our results also extend beyond the bipartite case previously studied. To further complement them, we also derive lower bounds for any deterministic reduction of ranking to binary classification, proving that randomization is necessary to achieve our reduction guarantees. 1

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 da”. 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

We consider the problem of finding a ranking of a set of elements that is “closest to ” a given set of input rankings of the elements; more precisely, we want to find a permutation that minimizes the Kendall-tau distance to the input rankings, where the Kendall-tau distance is defined as the sum over all input rankings of the number of pairs of elements that are in a different order in the input ranking than in the output ranking. If the input rankings are permutations, this problem is known as the Kemeny rank aggregation problem. This problem arises for example in building meta-search engines for Web search, aggregating viewers ’ rankings of movies, or giving recommendations to a user based on several different criteria, where we can think of having one ranking of the

Abstract. In the Π-Cluster Editing problem, one is given an undirected graph G, a density measure Π, and an integer k ≥ 0, and needs to decide whether it is possible to transform G by editing (deleting and inserting) at most k edges into a dense cluster graph. Herein, a dense cluster graph is a graph in which every connected component K = (VK, EK) satisfies Π. The well-studied Cluster Editing problem is a special case of this problem with Π:=“being a clique”. In this work, we consider three other density measures that generalize cliques: 1) having at most s missing edges (s-defective cliques), 2) having average degree at least |VK | − s (average-s-plexes), and 3) having average degree at least µ · (|VK | − 1) (µ-cliques), where s and µ are a fixed integer and a fixed rational number, respectively. We first show that the Π-Cluster Editing problem is NP-complete for all three density measures. Then, we study the fixed-parameter tractability of the three clustering problems, showing that the first two problems are fixed-parameter tractable with respect to the parameter (s, k) and that the third problem is W[1]-hard with respect to the parameter k for 0 < µ < 1. 1

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Abstract. Label ranking is a complex prediction task where the goal is to map instances to a total order over a finite set of predefined labels. An interesting aspect of this problem is that it subsumes several supervised learning problems such as multiclass prediction, multilabel classification and hierarchical classification. Unsurpisingly, there exists a plethora of label ranking algorithms in the literature due, in part, to this versatile nature of the problem. In this paper, we survey these algorithms. 1