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25
Aggregating inconsistent information: ranking and clustering
, 2005
"... We address optimization problems in which we are given contradictory pieces of input information and the goal is to find a globally consistent solution that minimizes the extent of disagreement with the respective inputs. Specifically, the problems we address are rank aggregation, the feedback arc s ..."
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Cited by 158 (9 self)
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We address optimization problems in which we are given contradictory pieces of input information and the goal is to find a globally consistent solution that minimizes the extent of disagreement with the respective inputs. Specifically, the problems we address are rank aggregation, the feedback arc set problem on tournaments, and correlation and consensus clustering. We show that for all these problems (and various weighted versions of them), we can obtain improved approximation factors using essentially the same remarkably simple algorithm. Additionally, we almost settle a longstanding conjecture of BangJensen and Thomassen and show that unless NP⊆BPP, there is no polynomial time algorithm for the problem of minimum feedback arc set in tournaments. 1
FixedParameter Algorithms for Cluster Vertex Deletion
, 2008
"... We initiate the first systematic study of the NPhard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixedparameter algorithmics. In the unweighted case, one searches for a minimum number of vertex deletions to transform a graph into a collection of disjoint cliques. Th ..."
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Cited by 23 (11 self)
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We initiate the first systematic study of the NPhard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixedparameter 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 fixedparameter 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 fixedparameter algorithms for (weighted) Vertex Cover.
Deterministic pivoting algorithms for constrained ranking and Clustering Problems
, 2007
"... 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], ..."
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Cited by 22 (3 self)
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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 minimumcost 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
Going weighted: Parameterized algorithms for cluster editing
 Theoretical Computer Science
"... Abstract. The goal of the Cluster Editing problem is to make the fewest changes to the edge set of an input graph such that the resulting graph is a disjoint union of cliques. This problem is NPcomplete but recently, several parameterized algorithms have been proposed. In this paper we present a su ..."
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Cited by 19 (3 self)
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Abstract. The goal of the Cluster Editing problem is to make the fewest changes to the edge set of an input graph such that the resulting graph is a disjoint union of cliques. This problem is NPcomplete but recently, several parameterized algorithms have been proposed. In this paper we present a surprisingly simple branching strategy for Cluster Editing. We generalize the problem assuming that edge insertion and deletion costs are positive integers. We show that the resulting search tree has size O(1.82 k)foreditcostk, resulting in the currently fastest parameterized algorithm for this problem. We have implemented and evaluated our approach, and find that it outperforms other parametrized algorithms for the problem.
FixedParameter Algorithms for Kemeny Scores
"... 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 permutat ..."
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Cited by 16 (8 self)
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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 NPhard. We provide first, encouraging fixedparameter 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
Exact algorithms for cluster editing: Evaluation and experiments
 Algorithmica
"... Abstract. We present empirical results for the Cluster Editing problem using exact methods from fixedparameter algorithmics and linear programming. We investigate parameterindependent data reduction methods and find that effective preprocessing is possible if the number of edge modifications k is ..."
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Cited by 14 (1 self)
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Abstract. We present empirical results for the Cluster Editing problem using exact methods from fixedparameter algorithmics and linear programming. We investigate parameterindependent data reduction methods and find that effective preprocessing is possible if the number of edge modifications k is smaller than some multiple of V . Inparticular, combining parameterdependent data reduction with lower and upper bounds we can effectively reduce graphs satisfying k ≤ 25 V . In addition to the fastest known fixedparameter branching strategy for the problem, we investigate an integer linear program (ILP) formulation of the problem using a cutting plane approach. Our results indicate that both approaches are capable of solving large graphs with 1000 vertices and several thousand edge modifications. For the first time, complex and very large graphs such as biological instances allow for an exact solution, using a combination of the above techniques. 1
A more relaxed model for graphbased data clustering: splex editing
 In Proc. 5th AAIM, LNCS
, 2009
"... Abstract. We introduce the sPlex Editing problem generalizing the wellstudied Cluster Editing problem, both being NPhard and both being motivated by graphbased data clustering. Instead of transforming a given graph by a minimum number of edge modifications into a disjoint union of cliques (Clust ..."
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Cited by 11 (7 self)
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Abstract. We introduce the sPlex Editing problem generalizing the wellstudied Cluster Editing problem, both being NPhard and both being motivated by graphbased 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 sPlex Editing is now to transform a graph into a disjoint union of socalled splexes. Herein, an splex denotes a vertex set inducing a (sub)graph where every vertex has edges to all but at most s vertices in the splex. Cliques are 1plexes. The advantage of splexes for s ≥ 2 is that they allow to model a more relaxed cluster notion (splexes 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 socalled problem kernel), a forbidden subgraph characterization of splex cluster graphs, and a depthbounded 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
FixedParameter Algorithms for Kemeny Rankings
, 2009
"... 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 NPhard. W ..."
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Cited by 11 (5 self)
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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 NPhard. 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 KendallTau distance da”. We describe a fixedparameter 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
An Efficient Reduction of Ranking to Classification
, 2007
"... 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 tha ..."
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Cited by 10 (2 self)
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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 upperbound. 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
Label Ranking Algorithms: A Survey
"... 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 ..."
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Cited by 9 (0 self)
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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