Abstract
In this paper, we initiate the study of the problem of ordering objects from their pairwise comparison results when allowed to discard up to a certain number of objects as outliers. More specifically, we seek to find an ordering under the popular Kendall tau distance measure, i.e., minimizing the number of pairwise comparison results that are inconsistent with the ordering, with some outliers removed. The presence of outliers challenges the assumption that a global consistent ordering exists and obscures the measure. This problem does not admit a polynomial time algorithm unless NP \( \subseteq \) BPP, and therefore, we develop approximation algorithms with provable guarantees for all inputs. Our algorithms have running time and memory usage that are almost linear in the input size. Further, they are readily adaptable to run on massively parallel platforms such as MapReduce or Spark.
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Notes
- 1.
A directed graph \(G = (V, E)\) is called a tournament if it is complete and directed. In other words, for any pair \(u \ne v \in V\), either \((u, v) \in E\) or \((v, u) \in E\).
- 2.
More precisely, he considered a slightly more general version where each vertex may have a different cost when removed as an outlier.
- 3.
We show that our algorithm is (180, 180)-approximate, which can be improved arbitrarily close to (60, 60) if one is willing to accept a lower success probability. In contrast, Aboud’s algorithm can be adapted to be (18, 18)-approximate for FASTO; however as mentioned above, it uses considerably more memory and run time than ours.
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Acknowledgements
This work was supported in part by NSF grants CCF-1409130 and CCF-1617653.
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Im, S., Montazer Qaem, M. (2020). Fast and Parallelizable Ranking with Outliers from Pairwise Comparisons. In: Brefeld, U., Fromont, E., Hotho, A., Knobbe, A., Maathuis, M., Robardet, C. (eds) Machine Learning and Knowledge Discovery in Databases. ECML PKDD 2019. Lecture Notes in Computer Science(), vol 11906. Springer, Cham. https://doi.org/10.1007/978-3-030-46150-8_11
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