Conference Paper

How Similarity Helps to Efficiently Compute Kemeny Rankings

DOI: 10.1145/1558013.1558104 Conference: 8th International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS 2009), Budapest, Hungary, May 10-15, 2009, Volume 1
Source: DBLP


The computation of Kemeny rankings is central to many applications in the context of rank aggregation. Unfortu- nately, the problem is NP-hard. We show that the Kemeny score (and a corresponding Kemeny ranking) of an election can be computed efficiently whenever the average pairwise distance between two input votes is not too large. In other words, Kemeny Score is fixed-parameter tractable with respect to the parameter "average pairwise Kendall-Tau dis- tance da". We describe a fixed-parameter algorithm with running time 16� d a� · poly. Moreover, we extend our stud- ies to the parameters "maximum range" and "average range" of positions a candidate takes in the input votes. Whereas Kemeny Score remains fixed-parameter tractable with re- spect to the parameter "maximum range", it becomes NP- complete in case of an average range value of two. This excludes fixed-parameter tractability with respect to the pa- rameter "average range" unless P=NP.

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Available from: Jiong Guo, Oct 09, 2015
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    • "16 kavg [5] 16 kmax [5] Figure 1: A summary of the running times proved in this paper and the best previous running times. Only the exponential terms are listed. "
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    ABSTRACT: We give improvements over fixed parameter tractable (FPT) algo- rithms to solve the Kemeny aggregation problem, where the task is to summarize a multi-set of preference lists, called votes, over a set of alternatives, called candidates, into a single preference list that has the minimum total τ-distance from the votes. The τ-distance between two preference lists is the number of pairs of candidates that are or- dered differently in the two lists. We study the problem for preference lists that are total orders. We develop algorithms of running times O�(1.403k t ), O�(5.823k t/m ) ≤ O�(5.823k avg ) and O�(4.829k max ) for the problem, ignoring the polynomial factors in the Onotation, where kt is the optimum total τ-distance, m is the number of votes, and kavg (resp, kmax) is the average (resp, maximum) over pairwise τ-distances of votes. Our algorithms improve the best previously known running times of O�(1.53kt) and O�(16kavg) ≤ O�(16kmax) (4, 5), which also implies an O�(164kt/m) running time. We also show how to enumerate all optimal solutions in O�(36kt/m) ≤ O�(36kavg) time.
    Parameterized and Exact Computation, 4th International Workshop, IWPEC 2009, Copenhagen, Denmark, September 10-11, 2009, Revised Selected Papers; 01/2009
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    ABSTRACT: There are different ways for an external agent to influence the outcome of an elec- tion. We concentrate on "control" by adding or deleting candidates. Our main focus is to investigate the parameterized complexity of various control problems for dif- ferent voting systems. To this end, we introduce natural digraph problems that may be of independent interest. They help in determining the parameterized complex- ity of control for different voting systems including Llull, Copeland, and plurality voting. Devising several parameterized reductions, we provide an overview of the parameterized complexity of the digraph and control problems with respect to nat- ural parameters such as adding/deleting only a bounded number of candidates or having only few voters.
    Combinatorial Optimization and Applications, Second International Conference, COCOA 2008, St. John's, NL, Canada, August 21-24, 2008. Proceedings; 01/2008
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