The problem of computing
a Kemeny optimal permutation
for a given collection of \(k\) full lists, for even integers \(k \geq 4\),
is NP-hard.

Minimum Feedback Edge Set

Minimum Feedback Edge Set

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Minimum Feedback Edge Set

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Minimum Feedback Edge Set

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If two rankings \(\sigma\) and \(\tau\)
are complete opposites,
then any ranking \(\rho\) will have
a total Kemeny distance of \({n \choose 2}\)

from both of them combined.

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This creates a neat

cancelation effect

& forces a minimum Kemeny score of

\(2{n \choose 2} + 2{m \choose 2} + 2m\)

on any ranking.

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This creates a neat

cancelation effect

& forces a minimum Kemeny score of

\(2{n \choose 2} + 2{m \choose 2} + 2m\)

on any ranking.

This creates a neat

cancelation effect

& forces a minimum Kemeny score of

\(2{n \choose 2} + 2{m \choose 2} + 2m\)

on any ranking.

+ 2k

if and only if G has a FES of size at most k.

final score \(\leq\)

(The cost of the backedges can be isolated.)

KRA Hardness

By Neeldhara Misra