On the hardness of maximum rank aggregation problems

The rank aggregation problem consists in finding a consensus ranking on a set of alternatives, based on the preferences of individual voters. The alternatives are expressed by permutations, whose pairwise distance can be measured in many ways.In this work we study a collection of distances, including the Kendall tau, Spearman footrule, Minkowski, Cayley, Hamming, Ulam, and related edit distances. Unlike the common median by summation, we compute the consensus against the maximum. The maximum consensus attempts to minimize the discrimination against any voter and is a smallest enclosing ball or center problem.We provide a general schema via local permutations for the NP-hardness of the maximum rank aggregation problems under all distances which satisfy some general requirements. This unifies former NP-hardness results for some distances and lays the ground for further ones. In particular, we establish a dichotomy for rank aggregation problems under the Spearman footrule and Minkowski distances: The median version is solvable in polynomial time whereas the maximum version is NP-hard. Moreover, we show that the maximum rank aggregation problem is 2-approximable under any pseudometric and fixed-parameter tractable under the Kendall tau, Hamming, and Minkowski distances, where again a general schema via modification sets applies.

[1]  V. Y. Popov,et al.  Multiple genome rearrangement by swaps and by element duplications , 2007, Theor. Comput. Sci..

[2]  Xiaotie Deng,et al.  On the complexity of crossings in permutations , 2009, Discret. Math..

[3]  B.J. Oommen,et al.  Pattern recognition of strings with substitutions, insertions, deletions and generalized transpositions , 1997, Pattern Recognit..

[4]  D. Critchlow.  Metric Methods for Analyzing Partially Ranked Data , 1986 .

[5]  Vladimir I. Levenshtein,et al.  Binary codes capable of correcting deletions, insertions, and reversals , 1965 .

[6]  Rolf Niedermeier,et al.  Invitation to Fixed-Parameter Algorithms , 2006 .

[7]  Nir Ailon,et al.  Aggregation of Partial Rankings, p-Ratings and Top-m Lists , 2007, SODA '07.

[8]  Dan Gusfield,et al.  Algorithms on strings , 1997 .

[9]  W. Knight.  A Computer Method for Calculating Kendall's Tau with Ungrouped Data , 1966 .

[10]  Christian Bachmaier,et al.  On Maximum Rank Aggregation Problems , 2013, IWOCA.

[11]  Michael R. Fellows,et al.  Review of: Fundamentals of Parameterized Complexity by Rodney G. Downey and Michael R. Fellows , 2015, SIGA.

[12]  Ami Litman,et al.  On covering problems of codes , 1997, Theory of Computing Systems.

[13]  Patricia Rose Gomes de Melo Viol Martins,et al.  MATHEMATICS WITHOUT NUMBERS: AN INTRODUCTION TO THE STUDY OF LOGIC , 2015 .

[14]  Marek Karpinski,et al.  Faster Algorithms for Feedback Arc Set Tournament, Kemeny Rank Aggregation and Betweenness Tournament , 2010, ISAAC.

[15]  Tayuan Huang,et al.  Metrics on Permutations, a Survey , 2004 .

[16]  Ely Porat,et al.  Swap and Mismatch Edit Distance , 2004, ESA.

[17]  Franz-Josef Brandenburg,et al.  The Nearest Neighbor Spearman Footrule Distance for Bucket, Interval, and Partial Orders , 2011, FAW-AAIM.

[18]  Dan Gusfield,et al.  Algorithms on Strings, Trees, and Sequences - Computer Science and Computational Biology , 1997 .

[19]  Guillaume Fertin,et al.  Sorting by Transpositions Is Difficult , 2010, SIAM J. Discret. Math..

[20]  R. Graham,et al.  Spearman's Footrule as a Measure of Disarray , 1977 .

[21]  Rolf Niedermeier,et al.  Fixed-Parameter Algorithms for CLOSEST STRING and Related Problems , 2003, Algorithmica.

[22]  Claire Mathieu,et al.  Electronic Colloquium on Computational Complexity, Report No. 144 (2006) How to rank with few errors A PTAS for Weighted Feedback Arc Set on Tournaments , 2006 .

[23]  Michael Ian Shamos,et al.  Closest-point problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[24]  Moni Naor,et al.  Rank aggregation methods for the Web , 2001, WWW '01.

[25]  Guillaume Fertin,et al.  Combinatorics of Genome Rearrangements , 2009, Computational molecular biology.

[26]  L. Bergroth,et al.  A survey of longest common subsequence algorithms , 2000, Proceedings Seventh International Symposium on String Processing and Information Retrieval. SPIRE 2000.

[27]  Alberto Caprara,et al.  Sorting Permutations by Reversals and Eulerian Cycle Decompositions , 1999, SIAM J. Discret. Math..

[28]  M. Trick,et al.  Voting schemes for which it can be difficult to tell who won the election , 1989 .

[29]  Franz-Josef Brandenburg,et al.  Comparing and Aggregating Partial Orders with Kendall Tau Distances , 2012, WALCOM.

[30]  Ronald Fagin.  Generalized first-order spectra, and polynomial. time recognizable sets , 1974 .

[31]  C. Y. Lee,et al.  Some properties of nonbinary error-correcting codes , 1958, IRE Trans. Inf. Theory.

[32]  Stefanie Seiler,et al.  Facility Layout And Location An Analytical Approach , 2016 .

[33]  Ronald Fagin,et al.  Comparing Partial Rankings , 2006, SIAM J. Discret. Math..

[34]  Leon F. McGinnis,et al.  Facility Layout and Location: An Analytical Approach , 1991 .

[35]  Rolf Niedermeier,et al.  Fixed-parameter algorithms for Kemeny rankings , 2009, Theor. Comput. Sci..

[36]  Ronald Fagin,et al.  Comparing top k lists , 2003, SODA '03.