Parameterized Approaches
Neeldhara Misra
to
KEMENY RANK AGGREGATION
IISc MSR Seminar Series
18 FEB 2022
(CSE @IIT Gandhinagar)
Plan
Input.
A collection of rankings
over a set of alternatives.
Input.
A collection of rankings
over a set of alternatives.
Input.
A collection of partial orders
over a set of alternatives.
Input.
A collection of weak orders
over a set of alternatives.
Input.
A collection of rankings
over a set of alternatives.
Input.
A collection of rankings
over a set of alternatives.
Output.
A single consensus ranking.
Input.
A collection of rankings
over a set of alternatives.
Output.
A single consensus ranking.
🍕 \(\succ\) 🍔 \(\succ\) 🍰 \(\succ\) 🍩 \(\succ\) 🌮
🍰 \(\succ\) 🍕 \(\succ\) 🌮 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
🌮 \(\succ\) 🍔 \(\succ\) 🍰 \(\succ\) 🍩 \(\succ\) 🍕
🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
🍕 \(\succ\) 🍔 \(\succ\) 🍰 \(\succ\) 🍩 \(\succ\) 🌮
🍰 \(\succ\) 🍕 \(\succ\) 🌮 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
🌮 \(\succ\) 🍔 \(\succ\) 🍰 \(\succ\) 🍩 \(\succ\) 🍕
🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
Input.
A collection of rankings
over a set of alternatives.
Output.
A single consensus ranking.
Input.
A collection of rankings
over a set of alternatives.
Output.
A single ranking that satisfies unanimity, yada yada.
Input.
A collection of rankings
over a set of alternatives.
Output.
A single ranking that satisfies unanimity, yada yada.
Input.
A collection of rankings
over a set of alternatives.
Output.
A single ranking that minimises
dissatisfication.
🍕 \(\succ\) 🍔 \(\succ\) 🍰 \(\succ\) 🍩 \(\succ\) 🌮
🍰 \(\succ\) 🍕 \(\succ\) 🌮 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
🌮 \(\succ\) 🍔 \(\succ\) 🍰 \(\succ\) 🍩 \(\succ\) 🍕
🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
🍕 \(\succ\) 🍔 \(\succ\) 🍰 \(\succ\) 🍩 \(\succ\) 🌮
🍰 \(\succ\) 🍕 \(\succ\) 🌮 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
🌮 \(\succ\) 🍔 \(\succ\) 🍰 \(\succ\) 🍩 \(\succ\) 🍕
🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
🍕 \(\succ\) 🍔 \(\succ\) 🍰 \(\succ\) 🍩 \(\succ\) 🌮
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
v/s
🍰 \(\succ\) 🍕 \(\succ\) 🌮 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
v/s
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
v/s
🌮 \(\succ\) 🍔 \(\succ\) 🍰 \(\succ\) 🍩 \(\succ\) 🍕
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
v/s
🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
v/s
🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
A notion of distance
between permutations would be
a useful measure of dissatisfaction.
🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
| 🍰 | 🍕 | 🍔 | 🍩 | 🌮 | |
|---|---|---|---|---|---|
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🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
| 🍰 | 🍕 | 🍔 | 🍩 | 🌮 | |
|---|---|---|---|---|---|
| 🍰 | ✔️ | ||||
| 🍕 | |||||
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| 🍩 | |||||
| 🌮 |
🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
| 🍰 | 🍕 | 🍔 | 🍩 | 🌮 | |
|---|---|---|---|---|---|
| 🍰 | ✔️ | ✔️ | |||
| 🍕 | |||||
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| 🌮 |
🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
| 🍰 | 🍕 | 🍔 | 🍩 | 🌮 | |
|---|---|---|---|---|---|
| 🍰 | ✔️ | ✔️ | ✔️ | ||
| 🍕 | |||||
| 🍔 | |||||
| 🍩 | |||||
| 🌮 |
🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
| 🍰 | 🍕 | 🍔 | 🍩 | 🌮 | |
|---|---|---|---|---|---|
| 🍰 | ✔️ | ✔️ | ✔️ | 🤦♀️ | |
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🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
| 🍰 | 🍕 | 🍔 | 🍩 | 🌮 | |
|---|---|---|---|---|---|
| 🍰 | ✔️ | ✔️ | ✔️ | 🤦♀️ | |
| 🍕 | ✔️ | ||||
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| 🍩 | |||||
| 🌮 |
🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
| 🍰 | 🍕 | 🍔 | 🍩 | 🌮 | |
|---|---|---|---|---|---|
| 🍰 | ✔️ | ✔️ | ✔️ | 🤦♀️ | |
| 🍕 | ✔️ | ✔️ | |||
| 🍔 | |||||
| 🍩 | |||||
| 🌮 |
🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
| 🍰 | 🍕 | 🍔 | 🍩 | 🌮 | |
|---|---|---|---|---|---|
| 🍰 | ✔️ | ✔️ | ✔️ | 🤦♀️ | |
| 🍕 | ✔️ | ✔️ | 🤷♀️ | ||
| 🍔 | |||||
| 🍩 | |||||
| 🌮 |
🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
| 🍰 | 🍕 | 🍔 | 🍩 | 🌮 | |
|---|---|---|---|---|---|
| 🍰 | ✔️ | ✔️ | ✔️ | 🤦♀️ | |
| 🍕 | ✔️ | ✔️ | 🤷♀️ | ||
| 🍔 | 😡 | ||||
| 🍩 | |||||
| 🌮 |
🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
| 🍰 | 🍕 | 🍔 | 🍩 | 🌮 | |
|---|---|---|---|---|---|
| 🍰 | ✔️ | ✔️ | ✔️ | 🤦♀️ | |
| 🍕 | ✔️ | ✔️ | 🤷♀️ | ||
| 🍔 | 😡 | 🥺 | |||
| 🍩 | |||||
| 🌮 |
🌮 \(\succ\) 🍰 \(\succ\) 🍕 \(\succ\) 🍩 \(\succ\) 🍔
🍰 \(\succ\) 🍕 \(\succ\) 🍔 \(\succ\) 🍩 \(\succ\) 🌮
| 🍰 | 🍕 | 🍔 | 🍩 | 🌮 | |
|---|---|---|---|---|---|
| 🍰 | ✔️ | ✔️ | ✔️ | 🤦♀️ | |
| 🍕 | ✔️ | ✔️ | 🤷♀️ | ||
| 🍔 | 😡 | 🥺 | |||
| 🍩 | 😭 | ||||
| 🌮 |
distance = #of disagreements
Input.
A collection of rankings
over a set of alternatives.
Output.
A single ranking that minimises
the total Kendall tau or bubble sort distance.
Bartholdi, Tovey and Tick, Social Choice and Welfare 1989,
Dwork, Kumar, Naor, and Sivakumar, WWW 2001
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
on Directed Graphs
Minimum Feedback Edge Set
on Directed Graphs
Minimum Feedback Edge Set
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Minimum Feedback Edge Set
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Minimum Feedback Edge Set
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Minimum Feedback Edge Set
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Minimum Feedback Edge Set
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Minimum Feedback Edge Set
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