Ranking Specific Sets of Objects Jan Maly, Stefan Woltran PPI17 @ - - PowerPoint PPT Presentation

Ranking Specific Sets of Objects

Jan Maly, Stefan Woltran

PPI17 @ BTW 2017, Stuttgart

March 7, 2017

Lifting rankings from objects to sets

Given

A set S A linear order < on S A family X ⊆ P(S) \ {∅} of nonempty subsets of S

The problem

Is there a “good” ranking ≺ on X ?

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 1

An example - Basics

S = {strawberry, vanilla, chocolate, lemon}

Linear order: strawberry < chocolate < vanilla < lemon

X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 2

The axiomatic approach

What is a “good” ranking?

The ranking should be based on the linear order < The ranking should be transitive, either reflexive or irreflexive, . . . “good” depends on the interpretation of X

Possible interpretations

Sets as final outcomes Opportunities Complete uncertainty

• etc. . .

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Axioms for ranking sets under complete uncertainty

Extension Rule

For all x, y ∈ S if {x}, {y} ∈ X , then

{x} ≺ {y} iff x < y Dominance

For all A ∈ X and all x ∈ S if A ∪ {x} ∈ X , then

y < x for all y ∈ A implies A ≺ A ∪ {x} x < y for all y ∈ A implies A ∪ {x} ≺ A

If X = P(S) \ {∅} and ≺ is transitive, the extension rule is implied by dominance.

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 4

An example - extension rule and dominance

S = {strawberry, vanilla, chocolate, lemon}

Linear order: strawberry < chocolate < vanilla < lemon

X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 5

An example - extension rule and dominance

S = {strawberry, vanilla, chocolate, lemon}

Linear order: strawberry < chocolate < vanilla < lemon

X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}}

Extension rule: {strawberry} ≺ {chocolate}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 5

An example - extension rule and dominance

S = {strawberry, vanilla, chocolate, lemon}

Linear order: strawberry < chocolate < vanilla < lemon

X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}}

Extension rule: {strawberry} ≺ {chocolate} Dominance:

{strawberry} ≺ {strawberry, vanilla} ≺ {strawberry, vanilla, lemon} {strawberry} ≺ {strawberry, lemon}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 5

Axioms for ranking sets under complete uncertainty

Independence

For all A, B ∈ X and for all x ∈ S \ (A ∪ B) if A ∪ {x}, B ∪ {x} ∈ X , then

A ≺ B implies A ∪ {x} B ∪ {x} Strict Independence

For all A, B ∈ X and for all x ∈ S \ (A ∪ B) if A ∪ {x}, B ∪ {x} ∈ X , then

A ≺ B implies A ∪ {x} ≺ B ∪ {x}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 6

An example - all axioms

S = {strawberry, vanilla, chocolate, lemon}

Linear order: strawberry < chocolate < vanilla < lemon

X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}}

Extension rule: {strawberry} ≺ {chocolate} Dominance:

{strawberry} ≺ {strawberry, vanilla} ≺ {strawberry, vanilla, lemon} {strawberry} ≺ {strawberry, lemon}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 7

An example - all axioms

S = {strawberry, vanilla, chocolate, lemon}

Linear order: strawberry < chocolate < vanilla < lemon

X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}}

Extension rule: {strawberry} ≺ {chocolate} Dominance:

{strawberry} ≺ {strawberry, vanilla} ≺ {strawberry, vanilla, lemon} {strawberry} ≺ {strawberry, lemon}

Independence: {strawberry, lemon} {strawberry, vanilla, lemon}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 7

An example - all axioms

S = {strawberry, vanilla, chocolate, lemon}

Linear order: strawberry < chocolate < vanilla < lemon

X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}}

Extension rule: {strawberry} ≺ {chocolate} Dominance:

{strawberry} ≺ {strawberry, vanilla} ≺ {strawberry, vanilla, lemon} {strawberry} ≺ {strawberry, lemon}

Independence: {strawberry, lemon} {strawberry, vanilla, lemon} Strict independence:

{strawberry, lemon} ≺ {strawberry, vanilla, lemon}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 7

Classic impossibility results

Kannai and Peleg (1984)

Assume X = P(S) \ {∅} and |S| ≥ 6, then there exists no order on X satisfying dominance and independence.

Barberà and Pattanaik (1984)

Assume X = P(S) \ {∅} and |S| ≥ 3, then there exists no binary relation

• n X satisfying dominance and strict independence.

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Proof of Barberà and Pattanaik

Assume S = {1, 2, 3} (1) {1} ≺ {1, 2} and (2) {2, 3} ≺ {3} by dominance

{1, 3} ≺ {1, 2, 3} by (1) and strict independence {1, 2, 3} ≺ {1, 3} by (2) and strict independence

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Proof of Kannai and Peleg

Observation: dominance and independence imply

A ∼ {max(A), min(A)}

Assume S = {1, 2, . . . , 6}

{3}?{2, 5}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg

Observation: dominance and independence imply

A ∼ {max(A), min(A)}

Assume S = {1, 2, . . . , 6}

{3} ≺ {2, 5}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg

Observation: dominance and independence imply

A ∼ {max(A), min(A)}

Assume S = {1, 2, . . . , 6}

{3} ≺ {2, 5} {3, 6} {2, 5, 6} by independence

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg

Observation: dominance and independence imply

A ∼ {max(A), min(A)}

Assume S = {1, 2, . . . , 6}

{3} ≺ {2, 5} {3, 6} {2, 5, 6} by independence {3, 4, 5, 6} {2, 3, 4, 5, 6} by observation

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg

Observation: dominance and independence imply

A ∼ {max(A), min(A)}

Assume S = {1, 2, . . . , 6}

{3} ≺ {2, 5} {3, 6} {2, 5, 6} by independence {3, 4, 5, 6} {2, 3, 4, 5, 6} by observation

This contradicts dominance!

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg

Observation: dominance and independence imply

A ∼ {max(A), min(A)}

Assume S = {1, 2, . . . , 6}

{2, 5} {3}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg

Observation: dominance and independence imply

A ∼ {max(A), min(A)}

Assume S = {1, 2, . . . , 6}

{2, 5} {3} ≺ {4}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg

Observation: dominance and independence imply

A ∼ {max(A), min(A)}

Assume S = {1, 2, . . . , 6}

{2, 5} ≺ {4}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg

Observation: dominance and independence imply

A ∼ {max(A), min(A)}

Assume S = {1, 2, . . . , 6}

{2, 5} ≺ {4} {1, 4} {1, 2, 5} by independence

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg

Observation: dominance and independence imply

A ∼ {max(A), min(A)}

Assume S = {1, 2, . . . , 6}

{2, 5} ≺ {4} {1, 4} {1, 2, 5} by independence {1, 2, 3, 4} {1, 2, 3, 4, 5} by observation

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Proof of Kannai and Peleg

Observation: dominance and independence imply

A ∼ {max(A), min(A)}

Assume S = {1, 2, . . . , 6}

{2, 5} ≺ {4} {1, 4} {1, 2, 5} by independence {1, 2, 3, 4} {1, 2, 3, 4, 5} by observation

This contradicts dominance!

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 10

Ditching the assumption X = P(S) \ {∅}

In many applications X is subject to constraints. There are families X = P(S) \ {∅} with |S| > 6 such that there is an

• rder on X satisfying dominance and (strict) independence.

It can be argued that dominance is too weak in the general case.

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An example - all axioms

S = {strawberry, vanilla, chocolate, lemon}

Linear order: strawberry < chocolate < vanilla < lemon

X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}}

Extension rule: {strawberry} ≺ {chocolate} Dominance:

{strawberry} ≺ {strawberry, vanilla} ≺ {strawberry, vanilla, lemon} {strawberry} ≺ {strawberry, lemon}

Independence: {strawberry, lemon} {strawberry, vanilla, lemon} Strict independence:

{strawberry, lemon} ≺ {strawberry, vanilla, lemon}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 12

An example - all axioms

S = {strawberry, vanilla, chocolate, lemon}

Linear order: strawberry < chocolate < vanilla < lemon

X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}}

Extension rule: {strawberry} ≺ {chocolate} Dominance:

{strawberry} ≺ {strawberry, vanilla} ≺ {strawberry, vanilla, lemon} {strawberry} ≺ {strawberry, lemon}

Independence: {strawberry, lemon} {strawberry, vanilla, lemon} Strict independence:

{strawberry, lemon} ≺ {strawberry, vanilla, lemon} {strawberry, vanilla}

?

≺ {strawberry, lemon}

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A strengthening of dominance

Solution: Define a stronger version of dominance

Maximal dominance

For all A, B ∈ X ,

(max(A) ≤ max(B) ∧ min(A) < min(B)) ∨ (max(A) < max(B) ∧ min(A) ≤ min(B)) → A ≺ B

Assuming X = P(S) \ {∅}, maximal dominance is implied by dominance and (strict) independence.

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 13

An example - maximal dominance

S = {strawberry, vanilla, chocolate, lemon}

Linear order: strawberry < chocolate < vanilla < lemon

X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 14

An example - maximal dominance

S = {strawberry, vanilla, chocolate, lemon}

Linear order: strawberry < chocolate < vanilla < lemon

X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}}

Maximal dominance:

{strawberry} ≺ {strawberry, vanilla} ≺ {strawberry, vanilla, lemon} {strawberry} ≺ {strawberry, lemon}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 14

An example - maximal dominance

S = {strawberry, vanilla, chocolate, lemon}

Linear order: strawberry < chocolate < vanilla < lemon

X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}}

Maximal dominance:

{strawberry} ≺ {strawberry, vanilla} ≺ {strawberry, vanilla, lemon} {strawberry} ≺ {strawberry, lemon} {strawberry} ≺ {chocolate}

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 14

An example - maximal dominance

S = {strawberry, vanilla, chocolate, lemon}

Linear order: strawberry < chocolate < vanilla < lemon

X = {{strawberry}, {chocolate}, {strawberry, vanilla}, {strawberry, vanilla, lemon}, {strawberry, lemon}}

Maximal dominance:

{strawberry} ≺ {strawberry, vanilla} ≺ {strawberry, vanilla, lemon} {strawberry} ≺ {strawberry, lemon} {strawberry} ≺ {chocolate} {strawberry, vanilla} ≺ {strawberry, lemon}

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The problems we treated

The Partial (Max)-Dominance-(Strict)-Independence Problem

Given a linearly ordered set S and a set X ⊆ P(S) \ {∅}, decide if there is a partial order/preorder on X satisfying (maximal) dominance and (strict) independence.

The (Max)-Dominance-(Strict)-Independence Problem

Given a linearly ordered set S and a set X ⊆ P(S) \ {∅}, decide if there is a (strict) total order on X satisfying (maximal) dominance and (strict) independence.

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The Partial (Max)-Dominance-(Strict)-Independence Problem

Theorem

The Partial (Max)-Dominance-Independence Problem is trivial. We can define a preorder satisfying maximal dominance and independence.

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The Partial (Max)-Dominance-(Strict)-Independence Problem

Theorem

The Partial (Max)-Dominance-Independence Problem is trivial. We can define a preorder satisfying maximal dominance and independence.

Theorem

The Partial (Max)-Dominance-Strict-Independence Problem is P-complete. We construct the minimal transitive relation satisfying (maximal) dominance and strict independence, then check if this relation is irreflexive. We can prove the P-hardness by a reduction from Horn-Sat.

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The Total (Max)-Dominance-(Strict)-Independence Problem

The (Max)-Dominance-(Strict)-Independence problem is NP-hard. This can be shown via a reduction from betweenness.

The Betweenness Problem

Given a finite set V = {v1, v2, . . . , vn} and a set of triples R ⊆ V3, find a strict total order on V such that a < b < c or a > b > c holds for all

(a, b, c) ∈ R.

The NP-hardness of betweenness was shown 1979 by Jaroslav Opatrny.

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The Total Max-Dominance-Strict-Independence Problem

Idea: Represent the elements v1, v2, . . . , vn of V by sets

V1, V2, . . . , Vn. Vi := {1, N} ∪ {i + 1, i + 2, . . . , N − i} for sufficiently large N.

All sets have the same maximal and minimal element. The second largest elements are decreasing and second smallest elements are increasing.

V1 V2 Vn

Figure: Sketch of the sets V1, V2 and Vn

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The Total Max-Dominance-Strict-Independence Problem

For the triple (a, b, c) ∈ R represented by the sets A, B, C we add the following sets, where k is unique for this triple: A \ {k}, B \ {k},

B \ {k + 1}, C \ {k + 1}, A \ {k + 2}, B \ {k + 2}, B \ {k + 3}, C \ {k + 3}

We want B \ {k + 1} ≺ A \ {k}, B \ {k} ≺ C \ {k + 1},

A \ {k + 2} ≺ B \ {k + 3} and C \ {k + 3} ≺ B \ {k + 2}

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The Total Max-Dominance-Strict-Independence Problem

A B C

B \ {k} A \ {k} B \ {k + 1} C \ {k + 1}

Figure: Family that forces that A ≺ B leads to B ≺ C

A B C

B \ {k + 2} A \ {k + 2} B \ {k + 3} C \ {k + 3}

Figure: Family that forces that A ≻ B leads to B ≻ C

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The Total Max-Dominance-Strict-Independence Problem

For the triple (a, b, c) ∈ R represented by the sets A, B, C we add the following sets, where k is unique for this triple: A \ {k}, B \ {k},

B \ {k + 1}, C \ {k + 1}, A \ {k + 2}, B \ {k + 2}, B \ {k + 3}, C \ {k + 3}

We want B \ {k + 1} ≺ A \ {k}, B \ {k} ≺ C \ {k + 1},

A \ {k + 2} ≺ B \ {k + 3} and C \ {k + 3} ≺ B \ {k + 2}

For example, we can force B \ {k + 1} ≺ A \ {k} by adding

A \ {k, k + 4}, B \ {k + 1, k + 4} and either A \ {1, k, k + 4}, B \ {1, k + 1, k + 4} or A \ {k, k + 4, N}, B \ {k + 1, k + 4, N}

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A summary of our results

Not total Total Dom + Ind always NP-complete Max Dom +Ind always NP-complete Dom + Strict Ind in P NP-complete Max Dom + Strict Ind in P NP-complete

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Future work

The complexity of the studied problems if X is given in a compact way. Characterize the sets X that have orders satisfying (maximal) dominance and (strict) independence. Study other axioms and interpretations.

Jan Maly, March 7, 2017 Ranking Specific Sets of Objects 23