Constraint Satisfaction and Graph Theory

Pavol Hell

SIAM DM, June 2008

Pavol Hell Constraint Satisfaction and Graph Theory

NP versus Colouring

k−colouring problemsProblems in NP

NP−complete

Constraint Satisfaction Problems

CSPAssign values to variables so that all constraints are satisfied

ExamplesSAT3-COL(x , y) ∈ {(1, 1), (2, 3)} and(x , z, w) ∈ {(2, 2, 1), (1, 3, 2), (2, 2, 2)}

ExamplesSAT

3-COL(x , y) ∈ {(1, 1), (2, 3)} and(x , z, w) ∈ {(2, 2, 1), (1, 3, 2), (2, 2, 2)}

ExamplesSAT3-COL

(x , y) ∈ {(1, 1), (2, 3)} and(x , z, w) ∈ {(2, 2, 1), (1, 3, 2), (2, 2, 2)}

ExamplesSAT3-COL(x , y) ∈ {(1, 1), (2, 3)} and(x , z, w) ∈ {(2, 2, 1), (1, 3, 2), (2, 2, 2)}

NP versus CSP

Problems in NP

NP−complete

Problems in CSP

Generalizing Colouring

Which Problems / ComplexitiesWhat is this problem?

Which Problems / Complexities

What is this problem?

Which Problems / ComplexitiesWhat is this problem?

= HOM(H)

Homomorphism problem HOM(H)A colouring of a graph G without the above pattern is exactly ahomomorphism to H

= HOM(H)

Homomorphism problem HOM(H)A colouring of a graph G without the above pattern is exactly ahomomorphism to H

Feder-VardiEach constraint satisfaction problem is polynomially equivalentto HOM(H) for some digraph H

NP versus CSP

HOM(H)

Problems in NP

NP−complete

Problems in CSP

Which ProblemClique Cutset Problem

Matrix partition problems (Feder-Hell-Motwani-Klein)

NP versus MPP

Matrix Partition Problems

NP−complete

Problems in NP

Suspicious Problems

Cameron-Eschen-Hoang-Sritharan

Stubborn problem

Feder-HellGiven a 3-edge-coloured Kn, colour the vertices without amonochromatic edge

Complexity ? (Feder-Hell-Kral-Sgall)

Suspicious Problems

Stubborn problem

Suspicious Problems

Stubborn problem

Which ProblemIs H partitionable into a triangle-free graph and a cograph?

Partition problems, generalized colouring problems,subcolouring problems, etc., Alekseev, Farrugia, Lozin,Broersma, Fomin, Nesetril, Woeginger, Ekim, de Werra,Stacho, MacGillivray, Yu, Hoang, Le, etc.

What Problem is This??

What Problem is This?

What Problem is This??

NP versus CSP

Which Problems / ComplexitiesHow general are these "pattern-forbidding colouring problems"?

Fagin + Feder-Vardi + Kun-Nesetril + Nesetril-TardifEvery problem in NP is polynomially equivalent to apattern-forbidding colouring problemEvery problem in CSP is polynomially equivalent to apattern-forbidding colouring problem with patterns onsingle edges (or... trees)

Some finite set of patterns corresponds to isomorphismcomplete problems. What does it look like?

NP versus CSP

Fagin + Feder-Vardi + Kun-Nesetril + Nesetril-Tardif

Every problem in NP is polynomially equivalent to apattern-forbidding colouring problemEvery problem in CSP is polynomially equivalent to apattern-forbidding colouring problem with patterns onsingle edges (or... trees)

NP versus CSP

Fagin + Feder-Vardi + Kun-Nesetril + Nesetril-TardifEvery problem in NP is polynomially equivalent to apattern-forbidding colouring problem

Every problem in CSP is polynomially equivalent to apattern-forbidding colouring problem with patterns onsingle edges (or... trees)

NP versus CSP

Fagin + Feder-Vardi + Kun-Nesetril + Nesetril-TardifEvery problem in NP is polynomially equivalent to apattern-forbidding colouring problemEvery problem in CSP is polynomially equivalent to apattern-forbidding colouring problem with patterns onsingle edges

(or... trees)

NP versus CSP

Problems in NP

NP−complete

Problems in CSP

NP versus CSP

NP−complete

with Single−edge Patterns

Pattern−forbidding ProblemsPattern−forbidding Problems

NP versus MPP

?. . .

NP−complete

with Two−vertex Patterns

Pattern−forbidding ProblemsPattern−forbidding Problems

NP−complete

NP versus MPP

Matrix Partition Problems

NP−complete

Problems in NP

NP versus CSP

Problems in NP

NP−complete

Problems in CSP

Polymorphisms

POL(H) for a digraph H

f : V (H)k → V (H) such thataibi ∈ E(H) ∀i =⇒ f (a1, a2, . . . , ak )f (b1, b2, . . . , bk ) ∈ E(H).

JeavonsIf POL(H) ⊆ POL(H ′), then HOM(H ′) reduces to HOM(H)

The more polymorphisms H has, the more likely is HOM(H) tobe polynomial

Polymorphisms

How small can POL(H) be?

Projective HPOL(H) consists only of all projections, composed withautomorphisms of H

The graph Kn, with n ≥ 3, is projective

ThereforeIf H is projective, then HOM(H) is NP-complete

Examples

Example Polymorphisms

Majority polymorphism:f (u, u, v) = f (u, v , u) = f (v , u, u) = uNear unanimity polymorphism:f (v , u, . . . , u) = · · · = f (u, u, . . . , v) = uWeak unanimity polymorphism:f (u, u, . . . , u) = u, f (v , u, . . . , u) = · · · = f (u, u, . . . , v)

Maltsev polymorphism: f (u, u, v) = f (v , u, u) = v

Examples

Example PolymorphismsMajority polymorphism:f (u, u, v) = f (u, v , u) = f (v , u, u) = u

Near unanimity polymorphism:f (v , u, . . . , u) = · · · = f (u, u, . . . , v) = uWeak unanimity polymorphism:f (u, u, . . . , u) = u, f (v , u, . . . , u) = · · · = f (u, u, . . . , v)

Examples

Example PolymorphismsMajority polymorphism:f (u, u, v) = f (u, v , u) = f (v , u, u) = uNear unanimity polymorphism:f (v , u, . . . , u) = · · · = f (u, u, . . . , v) = u

Weak unanimity polymorphism:f (u, u, . . . , u) = u, f (v , u, . . . , u) = · · · = f (u, u, . . . , v)

Examples

Example PolymorphismsMajority polymorphism:f (u, u, v) = f (u, v , u) = f (v , u, u) = uNear unanimity polymorphism:f (v , u, . . . , u) = · · · = f (u, u, . . . , v) = uWeak unanimity polymorphism:f (u, u, . . . , u) = u, f (v , u, . . . , u) = · · · = f (u, u, . . . , v)

Examples

Example PolymorphismsMajority polymorphism:f (u, u, v) = f (u, v , u) = f (v , u, u) = uNear unanimity polymorphism:f (v , u, . . . , u) = · · · = f (u, u, . . . , v) = uWeak unanimity polymorphism:f (u, u, . . . , u) = u, f (v , u, . . . , u) = · · · = f (u, u, . . . , v)

Polynomial Algorithms

Feder-Vardi, Jeavons, Bulatov, Bulatov-DalmauIf H has a near unanimity polymorphism, or a Maltsevpolymorphism, then the problem HOM(H) is in P

Polynomial Algorithms

Universal ToolAll known polynomial cases are attributable to some nicepolymorphism.

Feder-Vardi, Bulatov-Jeavons-Krokhin,Maroti-McKenzie, Nesetril-Siggers

HOM(H) is NP-complete if

H has no weak near unanimity polymorphismH is K3-partitionableH is block-projectivesome algebra in VAR((V (H),POL(H)) is projective,and all these conditions are equivalent

Conjecture

In all other cases HOM(H) can be solved in polynomial time

H has no weak near unanimity polymorphism

H is K3-partitionableH is block-projectivesome algebra in VAR((V (H),POL(H)) is projective,and all these conditions are equivalent

Conjecture

H has no weak near unanimity polymorphismH is K3-partitionable

H is block-projectivesome algebra in VAR((V (H),POL(H)) is projective,and all these conditions are equivalent

Conjecture

H has no weak near unanimity polymorphismH is K3-partitionableH is block-projective

some algebra in VAR((V (H),POL(H)) is projective,and all these conditions are equivalent

Conjecture

H has no weak near unanimity polymorphismH is K3-partitionableH is block-projectivesome algebra in VAR((V (H),POL(H)) is projective,

and all these conditions are equivalent

Conjecture

Feder-Vardi + Bulatov-Jeavons-Krokhin +Larose-Zadori

The Dichotomy Conjecture

The problem HOM(H) is

In P is H admits a weak near unanimity polymorphismNP-complete if H does not admit a weak near unanimitypolymorphism

In P is H admits a weak near unanimity polymorphism

NP-complete if H does not admit a weak near unanimitypolymorphism

In P is H admits a weak near unanimity polymorphismNP-complete if H does not admit a weak near unanimitypolymorphism

Bang-Jensen - Hell Conjecture (1990)

Example application:

Barto-Kozik-Niven 2008If H has neither sources nor sinks, then

HOM(H) is in P if H retracts to a cycleHOM(H) is NP-complete otherwise(there is no weak near unanimity polymorphism)

HOM(H) is in P if H retracts to a cycleHOM(H) is NP-complete otherwise

(there is no weak near unanimity polymorphism)

HOM(H) is in P if H retracts to a cycleHOM(H) is NP-complete otherwise(there is no weak near unanimity polymorphism)

Graphs with Polymorphisms

Reflexive GraphsThe following are equivalent

G has a majority polymorphism

G is a retract of a product of pathsG is cop-win and clique-Helly

Hell-Rival, Nowakowski-Rival, Pesch-PoguntkeFeder-Vardi - decidable in polynomial time

G has a majority polymorphismG is a retract of a product of paths

G is cop-win and clique-Helly

G has a majority polymorphismG is a retract of a product of pathsG is cop-win and clique-Helly

Hell-Rival, Nowakowski-Rival, Pesch-Poguntke

Feder-Vardi - decidable in polynomial time

Brewster-Feder-Hell-Huang-MacGillivray 2008

Reflexive GraphsAny chordal graph G admits a near-unanimitypolymorphism

A chordless cycle of length >3 does not admit anear-unanimity polymorphism

Reflexive GraphsAny chordal graph G admits a near-unanimitypolymorphismA chordless cycle of length >3 does not admit anear-unanimity polymorphism

Larose-Loten-Zadori 2005

Reflexive GraphsThe following statements are equivalent

G admits a near-unanimity polymorphism

G2 dismantles to the diagonalDecidable in polynomial time

G admits a near-unanimity polymorphismG2 dismantles to the diagonal

Decidable in polynomial time

G admits a near-unanimity polymorphismG2 dismantles to the diagonal

Decidable in polynomial time

G has a conservative near-unanimity polymorphism

(f (a, b, . . . ) ∈ {a, b, . . . })G is an interval graph

G has a conservative near-unanimity polymorphism(f (a, b, . . . ) ∈ {a, b, . . . })

G is an interval graph

G has a conservative near-unanimity polymorphism(f (a, b, . . . ) ∈ {a, b, . . . })G is an interval graph

General Digraphs

Hell-Nesetril-Zhu, Feder-Vardi, Dalmau-Krokhin-LaroseFor a connected H the following are equivalent

H admits a near-unanimity operation

H has bounded treewidth dualityHOM(H) can be expressed in DatalogRET(H) can be expressed by a first order formula

General Digraphs

H admits a near-unanimity operationH has bounded treewidth duality

HOM(H) can be expressed in DatalogRET(H) can be expressed by a first order formula

General Digraphs

H admits a near-unanimity operationH has bounded treewidth dualityHOM(H) can be expressed in Datalog

RET(H) can be expressed by a first order formula

General Digraphs

H admits a near-unanimity operationH has bounded treewidth dualityHOM(H) can be expressed in DatalogRET(H) can be expressed by a first order formula

Finite duality

Rossman, Atserias, Larose-Loten-ZadoriH has finite duality ⇐⇒

HOM(H) can be expressed by a first order formulasome retract G of H has the property that G ×Gdismantles to the diagonaland then CSP(H) is in P

Nesetril-TardifIf H has finite duality then H has tree duality

Finite duality

Rossman, Atserias, Larose-Loten-ZadoriH has finite duality ⇐⇒HOM(H) can be expressed by a first order formula

some retract G of H has the property that G ×Gdismantles to the diagonaland then CSP(H) is in P

Finite duality

Rossman, Atserias, Larose-Loten-ZadoriH has finite duality ⇐⇒HOM(H) can be expressed by a first order formulasome retract G of H has the property that G ×Gdismantles to the diagonal

and then CSP(H) is in P

Finite duality

Rossman, Atserias, Larose-Loten-ZadoriH has finite duality ⇐⇒HOM(H) can be expressed by a first order formulasome retract G of H has the property that G ×Gdismantles to the diagonaland then CSP(H) is in P

Finite duality

Rossman, Atserias, Larose-Loten-ZadoriH has finite duality ⇐⇒HOM(H) can be expressed by a first order formulasome retract G of H has the property that G ×Gdismantles to the diagonaland then CSP(H) is in P

Constraint Satisfaction and Graph Theory · Constraint Satisfaction and Graph Theory Pavol Hell...

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