Constraint Satisfaction and Fair Multi-Objective ...ggreco/files/IJCAI13.pdf · 23rd International Conference on Artificial Intelligence Constraint Satisfaction and Fair Multi-Objective

23rd International Conference on Artificial Intelligence

Constraint Satisfaction and Fair Multi-Objective Optimization Problems:

Foundations, Complexity, and Islands of Tractability

Gianluigi Greco and Francesco Scarcello

Constraint Satisfaction Problems

• Distribute the goods/ presents to the kids

• Goods are indivisible

K1 K3

g1

K2

g2 g3 g4 g5 g6 g7


Variables

K1, K2, and K3

Domain

g1, g2 , …, g7

Constraints

K1 {g1, g2 , g3}

K2 {g2, g3 , g4 , g5}

K3 {g4, g5 , g6 , g7}

K1 K2, K2 K3



K1 K3

g1

K2

g2 g3 g4 g5 g6 g7




K1 K3

g1

K2

g2 g3 g4 g5 g6 g7

Solution

K1 → g2

K2 → g4

K3 → g6

Variables

K1, K2, and K3

Domain

g1, g2 , …, g7

Constraints

K1 {g1, g2 , g3}

K2 {g2, g3 , g4 , g5}

K3 {g4, g5 , g6 , g7}

K1 K2, K2 K3

Optimization Functions in CSPs



K1 K3

g1

K2

g2 g3 g4 g5 g6 g7

• Kids have preferences over the presents

Valuation Function: = +

(K1/g1) = 3

(K1/g2) = 10

…

Value of the Solution

(K1/g1) + (K2/g4) + (K3/g6) = 21

Optimal (MAX) Solution

Maximizes the social welfare

3

10

6 4

2 2 3

6

9 2

2

Multi-Objective Optimization

K1 K3

g1

K2

g2 g3 g4 g5 g6 g7

3

10

6 4

2 2 3

6

9 2

2

Different Valuations:

h is defined on Kh

Combination Strategies:


K1 K3

g1

K2

g2 g3 g4 g5 g6 g7

3

10

6 4

2 2 3

6

9 2

2


h is defined on Kh


MAX-SUM (social welfare)

10 2 9


K1 K3

g1

K2

g2 g3 g4 g5 g6 g7

3

10

6 4

2 2 3

6

9 2

2


h is defined on Kh



LEX

10 2 9

10 3 2


K1 K3

g1

K2

g2 g3 g4 g5 g6 g7

3

10

6 4

2 2 3

6

9 2

2


h is defined on Kh



LEX

PARETO

10 2 9

10 3 2

3 6 9 e.g.

Related Literature

K1 K3

g1

K2

g2 g3 g4 g5 g6 g7

3

10

6 4

2 2 3

6

9 2

2


h is defined on Kh



LEX

PARETO

10 2 9

10 3 2

3 6 9 e.g.

[Bistarelli et al., Rossi et al.]

[Freuder et al.]

[Torrens and Faltings]

Related Literature

K1 K3

g1

K2

g2 g3 g4 g5 g6 g7

3

10

6 4

2 2 3

6

9 2

2

Santa’s goal is to distribute presents in a way

that the least lucky kid is as happy as possible.

Related Literature

K1 K3

g1

K2

g2 g3 g4 g5 g6 g7

3

10

6 4

2 2 3

6

9 2

2


that the least lucky kid is as happy as possible. Social welfare = 19 (max 21)

Related Literature

K1 K3

g1

K2

g2 g3 g4 g5 g6 g7

3

10

6 4

2 2 3

6

9 2

2



FAIR OPTIMIZATION

MAX-MIN

[Snow and Freuder, Dubois and Fortemps, Bouveret and Lematre]

Social welfare = 19 (max 21)

Related Literature

K1 K3

g1

K2

g2 g3 g4 g5 g6 g7

3

10

6 4

2 2 3

6

9 2

2



FAIR OPTIMIZATION

MAX-MIN

[Snow and Freuder, Dubois and Fortemps, Bouveret and Lematre]

Social welfare = 19 (max 21)

Limited expressiveness • functions on one variable/constraint

No complexity analysis

Overview

Model

Complexity

Decomposition

Methods

The Model

is a set of valuation functions

is such that

with

is a commutative, associative, and closed binary operator

The Model


is such that

with


10 2 9

10 3 2

3 6 9

(possible solutions)

4 6 6

4 6 9

…

…

…

The Model


is such that

with


10 2 9

10 3 2

3 6 9


4 6 6

4 6 9

…

…

…

The Model


is such that

with


10 2 9

10 3 2

3 6 9


4 6 6

4 6 9

…

…

…

lex

The Model


is such that

with


10 2 9

10 3 2

3 6 9


4 6 6

4 6 9

…

…

…

lex

Overview

Model

Complexity

Decomposition

Methods

Complexity of (LEX)MAX-MIN Solutions

Constraint satisfaction is NP-hard

Even without optimization functions…

Tractable classes of CSPs Based on the values in the constraint relations

Based on the structure of the constraint scopes





Based on the structure of the constraint scopes Treewidth [Dechter & Pearl]






K1 K2

K1

K3 K2

K2 K3






K1 K2

further variable/constraints K1

K3 K2

K2 K3

K4


Vertices correspond to the hyperedges

Each variable induces a connected subtree

K1 K2


K3 K2

K2 K3

K4

{K1,K2,K3}

{K1,K2} {K2,K3}

{K2,K3,K4}




K1 K2


K3 K2

K2 K3

K4

{K1,K2,K3}

{K1,K2} {K2,K3}

{K2,K3,K4}




K1 K2


K3 K2

K2 K3

K4

{K1,K2,K3}

{K1,K2} {K2,K3}

{K2,K3,K4}

Complexity of Acyclic Instances

Restrictions on


Restrictions on

in P

[Gottlob et al.]


Restrictions on

in P

[Gottlob et al.]

in P

in P


Restrictions on

in P

[Gottlob et al.]

in P

in P in P

Dynamic programming


Restrictions on

in P

[Gottlob et al.]

in P

in P in P

Dynamic programming

in P


Restrictions on

in P weakly NP-hard

[Gottlob et al.]

• Reduction from «Partition»

• Pseudo-polynomial

in P

in P in P

Dynamic programming

in P


Restrictions on

in P weakly NP-hard

[Gottlob et al.]



NP-hard in P

in P

Reduction from «Set Packing»

in P

Dynamic programming

in P


Restrictions on

in P weakly NP-hard

[Gottlob et al.]



NP-hard in P

in P


NP-hard

in P

Dynamic programming

in P


Restrictions on

in P weakly NP-hard

[Gottlob et al.]



NP-hard in P

in P


NP-hard

in P

Dynamic programming

in P a novel machinery is needed

Overview

Model

Complexity

Decomposition

Methods

Key Ideas

Guards for Valuation Functions

Decomposition Methods


A set of variables W is a guard for a valuation function if

separates the hypergraph in components where its domain

variables do not occur together with any variable occurring in

other valuation functions

W


W defines 3 components

C1

C2

C3







C1

C2

C3







C1

C2

C3

covers X1, which is in the domain of







C1

C2

C3


C2 (and C1) does not contain variables in the domain of and







C1

C2

C3


C2 (and C1) does not contain variables in the domain of and

is a guard for ; in fact, it is also a guard for the other functions

Key Ideas




Common Ideas

Generalize the notion of graph or hypergraph acyclicity

Associate a width to each instance, expressing its degree of cyclicity

Polynomial time algorithms for bounded-width CSP instances, running

in O(n w+1· logn)

Bounded-width CSP instances can be recognized in polynomial time

Bounded-width decompositions can be computed in polynomial time

Noticeable Examples

Tree decompositions

(Generalized) Hypertree decompositions

Generalized Hypertree Decompositions

),','(),',()',',,,(

)','()','()',',(),(

),(),',()',',',,(),,',,(

FXBqFXBpYXYXJj

ZYhZXgZFFfZYe

ZXdZCCcFCYYSbFCXXSaans

a(S,X,X’,C,F), b(S,Y,Y’,C’,F’)

j(J,X,Y,X’,Y’)

j(_,X,Y,_,_), c(C,C’,Z) j(_,_,_,X’,Y’), f(F,F’,Z’)

d(X,Z) e(Y,Z) h(Y’,Z’) g(X’,Z’), f(F,_,Z’)

p(B,X’,F) q(B’,X’,F)

Basic Conditions(1/2)


j(J,X,Y,X’,Y’)

j(_,X,Y,_,_), c(C,C’,Z) j(_,_,_,X’,Y’), f(F,F’,Z’)



• We group edges



j(J,X,Y,X’,Y’)

j(_,X,Y,_,_), c(C,C’,Z) j(_,_,_,X’,Y’), f(F,F’,Z’)



• Edges can partially be used

j(_,_,_,X’,Y’), f(F,F’,Z’)

Basic Conditions(2/2)

Connectdness Condition


j(J,X,Y,X’,Y’)

j(_,X,Y,_,_), c(C,C’,Z) j(_,_,_,X’,Y’), f(F,F’,Z’)



Hypertree Decompositions (HTD)


j(J,X,Y,X’,Y’)

j(_,X,Y,_,_), c(C,C’,Z) j(_,_,_,X’,Y’), f(F,F’,Z’)



Does not appear in

the subtrees rooted at v

J X Y

HTD = Generalized HTD +Special Condition

Each variable not used

at some vertex v

Key Ideas



Decomposition Methods and Guards

Input CSP A1 An

set of equivalent acyclic instances


An instance is guarded via the given method if there is an output

Ai such that each valution function is guarded by some hyperedge

Input CSP A1 An



An instance is guarded via the given method if there is an output

Ai such that each valution function is guarded by some hyperedge

Input CSP A1 An


is guarded via hypertree decomposition (width k=3)

Main Results

in P weakly NP-hard

NP-hard in P

in P

NP-hard

in P in P

Main Results

in P weakly NP-hard

NP-hard in P

in P

NP-hard

in P in P

in P, if guarded via

Proof Idea


guard for

variables in the domain of

Proof Idea


guard for


X1

Proof Idea


guard for


X1

solutions with optimal values, computed via dynamic programming

Proof Idea


guard for


X1

solutions with optimal values, computed via dynamic programming

X1 Xn

acyclic instance, with 1 function over n variables

Main Results

in P weakly NP-hard

NP-hard in P

in P

NP-hard

in P in P


Main Results

in P weakly NP-hard

NP-hard in P

in P

NP-hard

in P in P

(acyclic) instances of this kind are always guarded via width: h x k+1


Main Results

in P weakly NP-hard

NP-hard in P

in P

NP-hard

in P in P

in P

(acyclic) instances of this kind are always guarded via width: h x k+1


Overview

Model

Complexity

Decomposition

Methods

Documents

Constraint Satisfaction and Fair Multi-Objective ...ggreco/files/IJCAI13.pdf · 23rd International Conference on Artificial Intelligence Constraint Satisfaction and Fair Multi-Objective