How to rank with few errors

This paper presents an efficient preference-based ranking algorithm running in two stages. In the first stage, the algorithm learns a preference function defined over pairs, as in a standard binary classification problem. In the second stage, it makes use of that preference function to produce an accurate ranking, thereby reducing the learning problem of ranking to binary classification. This reduction is based on the familiar QuickSort and guarantees an expected pairwise misranking loss of at most twice that of the binary classifier derived in the first stage. Furthermore, in the important special case of bipartite ranking, the factor of two in loss is reduced to one. This improved bound also applies to the regret achieved by our ranking and that of the binary classifier obtained.Our algorithm is randomized, but we prove a lower bound for any deterministic reduction of ranking to binary classification showing that randomization is necessary to achieve our guarantees. This, and a recent result by Balcan et al., who show a regret bound of two for a deterministic algorithm in the bipartite case, suggest a trade-off between achieving low regret and determinism in this context.Our reduction also admits an improved running time guarantee with respect to that deterministic algorithm. In particular, the number of calls to the preference function in the reduction is improved from (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 algorithm is thus practical for realistic applications where the number of points to rank exceeds several thousand.

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“…Thus, this work also adds a significant contribution to Ailon et al (2005), Ailon (2007) and Kenyon-Mathieu and Schudy (2007).…”

Section: )mentioning

confidence: 83%

Preference-based learning to rank

Ailon

Mohri

2010

Mach Learn

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“…Fernau et al [8] reduced this problem to weighted FAST (feedback arc sets in tournaments) and, using the algorithm of Alon, Lokshtanov, and Saurabh [2] for weighted FAST, gave a subexponential time algorithm that runs in 2 O( √ k log k) + n O (1) time. This reduction also gave a PTAS using the algorithm of Kenyon-Mathieu and Schudy [13]. Karpinski and Schudy [12] considered a different version of weighted FAST proposed in [1], which imposes certain restrictions called probability constraints on the instances, and gave a faster algorithm that runs in 2 O( √ OPT) + n O (1) time where OPT is the cost of an optimal solution.…”

Section: Oscm (One-sided Crossing Minimization)mentioning

confidence: 99%

“…Fernau et al [8] gave a PTAS for OSCM using the PTAS of Kenyon-Mathieu and Schudy [13] for weighted FAST. It would be interesting to know whether our approach can be used to develop a PTAS.…”

Section: Future Workmentioning

confidence: 99%

A Fast and Simple Subexponential Fixed Parameter Algorithm for One-Sided Crossing Minimization

Kobayashi

Tamaki

2014

Algorithmica

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We give a subexponential fixed parameter algorithm for one-sided crossing minimization. It runs in O(k2where n is the number of vertices of the given graph and parameter k is the number of crossings. The exponent of O( √ k) in this bound is asymptotically optimal assuming the Exponential Time Hypothesis and the previously best known algorithm runs in 2 O( √ k log k) + n O(1) time. We achieve this significant improvement by the use of a certain interval graph naturally associated with the problem instance and a simple dynamic program on this interval graph. The linear dependency on n is also achieved through the use of this interval graph.

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“…The Kemeny score can be approximated to a factor of 8/5 by a deterministic algorithm [21] and to a factor of 11/7 by a randomized algorithm [2]. Recently, a polynomial-time approximation scheme (PTAS) for Kemeny Score has been developed [15]. However, its running time is impractical.…”

Section: Kemeny Rankingsmentioning

confidence: 99%

Average Parameterization and Partial Kernelization for Computing Medians

Betzler

Guo

Komusiewicz

et al. 2010

LATIN 2010: Theoretical Informatics

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Abstract. We propose an effective polynomial-time preprocessing strategy for intractable median problems. Developing a new methodological framework, we show that if the input instances of generally intractable problems exhibit a sufficiently high degree of similarity between each other on average, then there are efficient exact solving algorithms. In other words, we show that the median problems Swap Median Permutation, Consensus Clustering, Kemeny Score, and Kemeny Tie Score all are fixed-parameter tractable with respect to the parameter "average distance between input objects". To this end, we develop the new concept of "partial kernelization" and identify interesting polynomial-time solvable special cases for the considered problems.

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How to rank with few errors

Abstract: We present a polynomial time approximation scheme (PTAS) for the minimum feedback arc set problem on tournaments. A simple weighted generalization gives a PTAS for Kemeny-Young rank aggregation.

Cited by 153 publications

References 39 publications

Preference-based learning to rank

Preference-based learning to rank

A Fast and Simple Subexponential Fixed Parameter Algorithm for One-Sided Crossing Minimization

Average Parameterization and Partial Kernelization for Computing Medians

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