Fixed-Parameter Algorithms for Kemeny Scores

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 NP-hard. We provide first, encouraging fixed-parameter tractability results for computing optimal scores (that is, the overall distance of an optimal consensus permutation). Our fixed-parameter 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]  Jianer Chen,et al.  Directed Feedback Vertex Set Problem is FPT , 2007, Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs.

[2]  Barry O'Sullivan,et al.  A fixed-parameter algorithm for the directed feedback vertex set problem , 2008, JACM.

[3]  Dimitrios M. Thilikos,et al.  Invitation to fixed-parameter algorithms , 2007, Comput. Sci. Rev..

[4]  M. Truchon,et al.  An Extension of the Concordet Criterion and Kemeny Orders , 1998 .

[5]  Michael R. Fellows,et al.  Parameterized Complexity , 1998 .

[6]  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 .

[7]  Anders Yeo,et al.  The Minimum Feedback Arc Set Problem is NP-Hard for Tournaments , 2006, Combinatorics, Probability and Computing.

[8]  Nir Ailon,et al.  Aggregating inconsistent information: Ranking and clustering , 2008 .

[9]  Saket Saurabh,et al.  Improved fixed parameter tractable algorithms for two "edge" problems: MAXCUT and MAXDAG , 2007, Inf. Process. Lett..

[10]  Jayant Kalagnanam,et al.  A Computational Study of the Kemeny Rule for Preference Aggregation , 2004, AAAI.

[11]  Rolf Niedermeier,et al.  A general method to speed up fixed-parameter-tractable algorithms , 2000, Inf. Process. Lett..

[12]  David P. Williamson,et al.  Deterministic Algorithms for Rank Aggregation and Other Ranking and Clustering Problems , 2007, WAOA.

[13]  Vincent Conitzer,et al.  Improved Bounds for Computing Kemeny Rankings , 2006, AAAI.

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

[15]  Edith Hemaspaandra,et al.  The complexity of Kemeny elections , 2005, Theor. Comput. Sci..

[16]  V. Rich.  Personal communication , 1989, Nature.

[17]  Vincent Conitzer,et al.  Computing Slater Rankings Using Similarities among Candidates , 2006, AAAI.

[18]  Jörg Flum,et al.  Parameterized Complexity Theory (Texts in Theoretical Computer Science. An EATCS Series) , 2006 .

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

[20]  Noga Alon,et al.  Ranking Tournaments , 2006, SIAM J. Discret. Math..

[21]  Jörg Flum,et al.  Parameterized Complexity Theory , 2006, Texts in Theoretical Computer Science. An EATCS Series.