Parameterized Enumeration of (Locally-) Optimal Aggregations

  • Naomi Nishimura
  • Narges Simjour
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)


We present a parameterized enumeration algorithm for Kemeny Rank Aggregation, the problem of determining an optimal aggregation, a total order that is at minimum total τ-distance (k t ) from the input multi-set of m total orders (votes) over a set of alternatives (candidates), where the τ-distance between two total orders is the number of pairs of candidates ordered differently. Our \(O^*(4^{k_t\over m})\)-time algorithm constitutes a significant improvement over the previous \(O^*(36^{k_t\over m})\) upper bound.

The analysis of our algorithm relies on the notion of locally-optimal aggregations, total orders whose total τ-distances from the votes do not decrease by any single swap of two candidates adjacent in the ordering. As a consequence of our approach, we provide not only an upper bound of \(4^{k_t\over m}\) on the number of optimal aggregations, but also the first parameterized bound, \(4^{k_t\over m}\), on the number of locally-optimal aggregations, and demonstrate that it is tight. Furthermore, since our results rely on a known relation to Weighted Directed Feedback Arc Set, we obtain new results for this problem along the way.


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Naomi Nishimura
    • 1
  • Narges Simjour
    • 1
  1. 1.Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

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