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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    • "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.
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