Extensión del concepto de utopía para el problema de la agregación de rankings sin empates

The use of rankings and how to aggregate or summarize them has received increasing attention in various fields: bibliometrics, web search, data mining, statistics, educational quality, and computational biology. For the Optimal Bucket Order Problem, the concept of Utopian Matrix was recently introduced: an ideal and not necessarily feasible solution with an unsurpassed quality for the feasible solutions of the problem. This work proposes an extension of the notion of Utopian Matrix to the Rank Aggregation Problem in which ties are not allowed between elements in the output ranking. Beyond the extension that is direct, the work focuses on studying its usefulness as an idealization or super optimal solution. As the Rank Aggregation Problem can be solved exactly based on its definition as an Integer Linear Programming Problem, an experimental study is presented where it is analyzed the relationship that exists between utopian (and anti utopian) values and the optimal solution in several instances solved by using the open source software SCIP. Among the 47 instances analyzed, in 19 the Utopian Value turned out to be equal to the optimal value (40.43 % feasibility) and in 18 the Anti Utopian Value also turned out to be feasible (38.00 %). This experimental study demonstrates the usefulness of utopian and anti utopian values to be considered as extreme values in the Rank Aggregation Problem, thus being able to find higher and lower bounds for optimization very quickly.El uso de los rankings y la forma de agregarlos o resumirlos ha recibido una atención creciente en diversos campos: bibliometría, búsquedas web, minería de datos, estadística, calidad educativa y biología computacional. Para el Problema de Ordenamiento Óptimo con empates fue introducido recientemente el concepto de Matriz Utópica: una solución ideal y no necesariamente factible con una calidad insuperable para las soluciones factibles del problema. Este trabajo propone una extensión de la noción de Matriz Utópica para el Problema de Agregación de Rankings en que no se permiten empates entre elementos en el ranking de salida. Más allá de la extensión que es directa, el trabajo se centra en estudiar su valor como idealización o solución súper óptima. Como el Problema de Agregación de Rankings puede resolverse de forma exacta a partir de su definición como Problema de Programación Lineal Entera, se presenta un estudio experimental donde se analiza la relación que existe entre los valores utópicos (y anti utópicos) y la solución óptima en instancias resueltas con la ayuda del software de código abierto SCIP. Entre las 47 instancias analizadas, en 19 el Valor Utópico resultó ser igual al valor óptimo (40,43 % de factibilidad) y en 18 el Valor Anti Utópico también resultó ser factible (38,00 %). Este estudio experimental demuestra la utilidad de los valores utópicos y anti utópicos para ser considerados como valores extremos en el Problema de Agregación de Rankings, pudiendo así encontrase muy rápidamente cotas superiores e inferiores para la optimización

Approaching rank aggregation problems by using evolution strategies: The case of the optimal bucket order problem

The optimal bucket order problem consists in obtaining a complete consensus ranking (ties are allowed) from a matrix of preferences (possibly obtained from a database of rankings). In this paper, we tackle this problem by using evolution strategies. We designed specific mutation operators which are able to modify the inner structure of the buckets, which introduces more diversity into the search process. We also study different initialization methods and strategies for the generation of the population of descendants. The proposed evolution strategies are tested using a benchmark of 52 databases and compared with the current state-of-the-art algorithm LIA. We carry out a standard machine learning statistical analysis procedure to identify a subset of outstanding configurations of the proposed evolution strategies. The study shows that the best evolution strategy improves upon the accuracy obtained by the standard greedy method (BPA) by 35%, and that of LIA by 12.5%

Discovering bucket orders from full rankings

Discovering a bucket order B from a collection of possibly noisy full rankings is a fundamental problem that relates to various applications involving rankings. Informally, a bucket order is a total order that allows "ties" between items in a bucket. A bucket order B can be viewed as a "representative" that summarizes a given set of full rankings {7 ₁, T₂, . . ., T_m}, or conversely B can be an "approximation" of some "ground truth" G where the rankings {T₁, T₂,.. ., T_m} are the "linear extensions" of G. Current work of finding bucket orders such as the dynamic programming algorithm is mainly developed from the "representative" perspective, which maximizes items' Intra-bucket similarity when forming a bucket. The underlying idea of maximizing intra-bucket similarity is realized via minimizing the sum of the deviations of median ranks within a bucket. In contrast, from the "approximation" perspective, since each observed full ranking T_i is simply a linear extension of the given "ground truth" bucket order G, items in a big bucket b in G are forced to have different median ranks, and as a result b will have a big sum of deviations. Thus, minimizing the sum of deviations may result in an undesirable scenario that big buckets are mostly decomposed into small ones. In this paper, we propose a novel heuristic called Abnormal Rank Gap to capture the inter-bucket dissimilarity for better bucket forming. In addition, we propose to use the "closeness" on multiple quantile ranks to determine if two items should be put into the same bucket. We develop a novel bucket order discovering method called the Bucket Gap algorithm. Our extensive experiments demonstrate that the Bucket Gap algorithm significantly outperforms the major related work, i.e., the Bucket Pivot algorithm. In particular, the error distance of the generated bucket order can be reduced by about 30% on a real paleontological dataset and the noise tolerance can be increased from 30% to 50% in the synthetic dataset. © Copyright 2008 ACM