Constraints, Symmetry, and Complexity Andrei Krokhin Andrei Krokhin - - PowerPoint PPT Presentation

Constraints, Symmetry, and Complexity Andrei Krokhin

Andrei Krokhin Constraints, Symmetry, and Complexity

Message

Q.: What kind of mathematical structure makes a problem easy or hard?

Andrei Krokhin Constraints, Symmetry, and Complexity

Message

Q.: What kind of mathematical structure makes a problem easy or hard?

The Constraint Satisfaction Problem paradigm

— General enough to include many interesting problems — Structured enough to make significant progress (on all problems within a class, not just a single problem)

Andrei Krokhin Constraints, Symmetry, and Complexity

Message

Q.: What kind of mathematical structure makes a problem easy or hard?

The Constraint Satisfaction Problem paradigm Main Achievement: better understanding of common structure in easy (or hard) problems

— focus on appropriate symmetries of solution spaces! — lack of symmetries ⇒ hardness — symmetries ⇒ efficient algorithms

Andrei Krokhin Constraints, Symmetry, and Complexity

Message

Q.: What kind of mathematical structure makes a problem easy or hard?

The Constraint Satisfaction Problem paradigm Main Achievement: better understanding of common structure in easy (or hard) problems

— focus on appropriate symmetries of solution spaces! — lack of symmetries ⇒ hardness — symmetries ⇒ efficient algorithms — symmetries of higher dimension/arity are important (not just auto- or endomorphisms) → universal algebra

Andrei Krokhin Constraints, Symmetry, and Complexity

Message

Q.: What kind of mathematical structure makes a problem easy or hard?

The Constraint Satisfaction Problem paradigm Main Achievement: better understanding of common structure in easy (or hard) problems

— focus on appropriate symmetries of solution spaces! — lack of symmetries ⇒ hardness — symmetries ⇒ efficient algorithms — symmetries of higher dimension/arity are important (not just auto- or endomorphisms) → universal algebra — useful way to measure/compare symmetries

Andrei Krokhin Constraints, Symmetry, and Complexity

Message

Q.: What kind of mathematical structure makes a problem easy or hard?

The Constraint Satisfaction Problem paradigm Main Achievement: better understanding of common structure in easy (or hard) problems

— focus on appropriate symmetries of solution spaces! — lack of symmetries ⇒ hardness — symmetries ⇒ efficient algorithms — symmetries of higher dimension/arity are important (not just auto- or endomorphisms) → universal algebra — useful way to measure/compare symmetries

Long term goal: go beyond CSPs

Andrei Krokhin Constraints, Symmetry, and Complexity

Constraint Satisfaction Problem (CSP)

Fix finite relational structure A = (A; R1, . . . , Rn) (aka constraint language) where each Ri ⊆ Aki

• r

Ri : Aki → {true, false}.

Andrei Krokhin Constraints, Symmetry, and Complexity

Constraint Satisfaction Problem (CSP)

Fix finite relational structure A = (A; R1, . . . , Rn) (aka constraint language) where each Ri ⊆ Aki

• r

Ri : Aki → {true, false}. Definition An instance of CSP(A) is a list of constraints over vars V , e.g. R1(x, y, z), R1(z, y, w), R2(z), R3(x, w), R3(y, y) where each Ri is from A. Question: Is there s : V → A satisfying all constraints?

Andrei Krokhin Constraints, Symmetry, and Complexity

Constraint Satisfaction Problem (CSP)

Fix finite relational structure A = (A; R1, . . . , Rn) (aka constraint language) where each Ri ⊆ Aki

• r

Ri : Aki → {true, false}. Definition An instance of CSP(A) is a list of constraints over vars V , e.g. R1(x, y, z), R1(z, y, w), R2(z), R3(x, w), R3(y, y) where each Ri is from A. Question: Is there s : V → A satisfying all constraints? Other variants, e.g.: infinite A (but the instance is still finite) nothing (or something other than relations) is fixed real-valued functions instead of relations (for optimisation)

Andrei Krokhin Constraints, Symmetry, and Complexity

Examples and a conjecture

k-Col: A = ([k]; {=})

Andrei Krokhin Constraints, Symmetry, and Complexity

Examples and a conjecture

k-Col: A = ([k]; {=}) k-NAE: A = ([k]; {(a, b, c) ∈ [k]3 | a = b ∨ a = c ∨ b = c})

— (essentially) k-colouring for 3-uniform hypergraphs

Andrei Krokhin Constraints, Symmetry, and Complexity

Examples and a conjecture

k-Col: A = ([k]; {=}) k-NAE: A = ([k]; {(a, b, c) ∈ [k]3 | a = b ∨ a = c ∨ b = c})

— (essentially) k-colouring for 3-uniform hypergraphs

3-Sat: A = ({0, 1}; (x ∨ y ∨ z), (x ∨ y ∨ ¬z), . . .) Horn 3-Sat: as above, each clause has ≤ 1 unneg var

Andrei Krokhin Constraints, Symmetry, and Complexity

Examples and a conjecture

k-Col: A = ([k]; {=}) k-NAE: A = ([k]; {(a, b, c) ∈ [k]3 | a = b ∨ a = c ∨ b = c})

— (essentially) k-colouring for 3-uniform hypergraphs

3-Sat: A = ({0, 1}; (x ∨ y ∨ z), (x ∨ y ∨ ¬z), . . .) Horn 3-Sat: as above, each clause has ≤ 1 unneg var 3-Linp: A = (Zp; x + y + z = 0, x + 2y + 3z = 7, . . .)

Andrei Krokhin Constraints, Symmetry, and Complexity

Examples and a conjecture

k-Col: A = ([k]; {=}) k-NAE: A = ([k]; {(a, b, c) ∈ [k]3 | a = b ∨ a = c ∨ b = c})

— (essentially) k-colouring for 3-uniform hypergraphs

3-Sat: A = ({0, 1}; (x ∨ y ∨ z), (x ∨ y ∨ ¬z), . . .) Horn 3-Sat: as above, each clause has ≤ 1 unneg var 3-Linp: A = (Zp; x + y + z = 0, x + 2y + 3z = 7, . . .) Unique Games-k: A = ([k]; {(a, π(a)) | a ∈ [k]}) where π runs through all permutations on [k].

Andrei Krokhin Constraints, Symmetry, and Complexity

Examples and a conjecture

k-Col: A = ([k]; {=}) k-NAE: A = ([k]; {(a, b, c) ∈ [k]3 | a = b ∨ a = c ∨ b = c})

— (essentially) k-colouring for 3-uniform hypergraphs

3-Sat: A = ({0, 1}; (x ∨ y ∨ z), (x ∨ y ∨ ¬z), . . .) Horn 3-Sat: as above, each clause has ≤ 1 unneg var 3-Linp: A = (Zp; x + y + z = 0, x + 2y + 3z = 7, . . .) Unique Games-k: A = ([k]; {(a, π(a)) | a ∈ [k]}) where π runs through all permutations on [k]. Conjecture (CSP Dichotomy Conjecture, Feder-Vardi’98) Every CSP(A) is either in P or NP-c.

Andrei Krokhin Constraints, Symmetry, and Complexity

Some algorithmic tasks about CSPs

Andrei Krokhin Constraints, Symmetry, and Complexity

Some algorithmic tasks about CSPs

Programme: Classify the complexity of these tasks wrt A

• 1. Decision CSP: Can all constraints be satisfied?

— Done [Bulatov’17, Zhuk’17]; finer classification — open

Andrei Krokhin Constraints, Symmetry, and Complexity

Some algorithmic tasks about CSPs

Programme: Classify the complexity of these tasks wrt A

• 1. Decision CSP: Can all constraints be satisfied?

— Done [Bulatov’17, Zhuk’17]; finer classification — open

• 2. Counting CSP: Count the number of solutions

— Done [Bulatov’08-13, Dyer, Richerby’10-13, Cai, Chen’12-17]

Andrei Krokhin Constraints, Symmetry, and Complexity

Some algorithmic tasks about CSPs

Programme: Classify the complexity of these tasks wrt A

• 1. Decision CSP: Can all constraints be satisfied?

— Done [Bulatov’17, Zhuk’17]; finer classification — open

• 2. Counting CSP: Count the number of solutions

— Done [Bulatov’08-13, Dyer, Richerby’10-13, Cai, Chen’12-17]

• 3. Max CSP: Find a map satisfying max number of constraints

— Done [Thapper, ˇ Zivn´ y’16], [Kolmogorov, AK, Rol´ ınek’17]

Andrei Krokhin Constraints, Symmetry, and Complexity

Some algorithmic tasks about CSPs

Programme: Classify the complexity of these tasks wrt A

• 1. Decision CSP: Can all constraints be satisfied?

— Done [Bulatov’17, Zhuk’17]; finer classification — open

• 2. Counting CSP: Count the number of solutions

— Done [Bulatov’08-13, Dyer, Richerby’10-13, Cai, Chen’12-17]

• 3. Max CSP: Find a map satisfying max number of constraints

— Done [Thapper, ˇ Zivn´ y’16], [Kolmogorov, AK, Rol´ ınek’17]

• 4. Approx Max CSP: Satisfy c × Opt number of constraints

— (Essentially) Done [Raghavendra’08]

Andrei Krokhin Constraints, Symmetry, and Complexity

Some algorithmic tasks about CSPs

Programme: Classify the complexity of these tasks wrt A

• 1. Decision CSP: Can all constraints be satisfied?

— Done [Bulatov’17, Zhuk’17]; finer classification — open

• 2. Counting CSP: Count the number of solutions

— Done [Bulatov’08-13, Dyer, Richerby’10-13, Cai, Chen’12-17]

• 3. Max CSP: Find a map satisfying max number of constraints

— Done [Thapper, ˇ Zivn´ y’16], [Kolmogorov, AK, Rol´ ınek’17]

• 4. Approx Max CSP: Satisfy c × Opt number of constraints

— (Essentially) Done [Raghavendra’08]

• 5. (α, β)-Approx Max CSP: assuming β fraction of constraints

can be satisfied, find a map satisfying ≥ α fraction.

— Strong results for some (α, β) and A, wide open in general

Andrei Krokhin Constraints, Symmetry, and Complexity

Some algorithmic tasks about CSPs

Programme: Classify the complexity of these tasks wrt A

• 1. Decision CSP: Can all constraints be satisfied?

— Done [Bulatov’17, Zhuk’17]; finer classification — open

• 2. Counting CSP: Count the number of solutions

— Done [Bulatov’08-13, Dyer, Richerby’10-13, Cai, Chen’12-17]

• 3. Max CSP: Find a map satisfying max number of constraints

— Done [Thapper, ˇ Zivn´ y’16], [Kolmogorov, AK, Rol´ ınek’17]

• 4. Approx Max CSP: Satisfy c × Opt number of constraints

— (Essentially) Done [Raghavendra’08]

• 5. (α, β)-Approx Max CSP: assuming β fraction of constraints

can be satisfied, find a map satisfying ≥ α fraction.

— Strong results for some (α, β) and A, wide open in general

• 6. Promise CSP: wide open — tomorrow

Andrei Krokhin Constraints, Symmetry, and Complexity

Algebraic approach to CSP (very high-level view) Relational structures

“can simulate”

. . . . . . . . . . . . . . .

Andrei Krokhin Constraints, Symmetry, and Complexity

Algebraic approach to CSP (very high-level view) Relational structures

“can simulate”

. . . . . . . . . . . . . . .

Strong enough polymorphisms (symmetries)

Andrei Krokhin Constraints, Symmetry, and Complexity

Algebraic approach to CSP (very high-level view) Relational structures

“can simulate”

3-Sat . . . . . . . . . . . . . . .

Strong enough polymorphisms (symmetries)

Andrei Krokhin Constraints, Symmetry, and Complexity

Simulation by example

Three (increasingly more general) levels of simulation:

Andrei Krokhin Constraints, Symmetry, and Complexity

Simulation by example

Three (increasingly more general) levels of simulation:

• 1. pp-definitions (= gadgets, same domain)

Andrei Krokhin Constraints, Symmetry, and Complexity

Simulation by example

Three (increasingly more general) levels of simulation:

• 1. pp-definitions (= gadgets, same domain) Ex.: Let

A1 = (A; R), R ternary, and A2 = (A; T, S) be s.t. T(x) = ∃z R(x, z, x), S(x, y) = ∃z R(x, y, z) ∧ R(y, y, x).

Andrei Krokhin Constraints, Symmetry, and Complexity

Simulation by example

Three (increasingly more general) levels of simulation:

• 1. pp-definitions (= gadgets, same domain) Ex.: Let

A1 = (A; R), R ternary, and A2 = (A; T, S) be s.t. T(x) = ∃z R(x, z, x), S(x, y) = ∃z R(x, y, z) ∧ R(y, y, x). Then an instance of CSP(A2), say T(y), S(x, y)

Andrei Krokhin Constraints, Symmetry, and Complexity

Simulation by example

Three (increasingly more general) levels of simulation:

• 1. pp-definitions (= gadgets, same domain) Ex.: Let

A1 = (A; R), R ternary, and A2 = (A; T, S) be s.t. T(x) = ∃z R(x, z, x), S(x, y) = ∃z R(x, y, z) ∧ R(y, y, x). Then an instance of CSP(A2), say T(y), S(x, y) can be re-written as an instance of CSP(A1) R(y, z, y), R(x, y, w), R(y, y, x)

Andrei Krokhin Constraints, Symmetry, and Complexity

Simulation by example

Three (increasingly more general) levels of simulation:

• 1. pp-definitions (= gadgets, same domain) Ex.: Let

A1 = (A; R), R ternary, and A2 = (A; T, S) be s.t. T(x) = ∃z R(x, z, x), S(x, y) = ∃z R(x, y, z) ∧ R(y, y, x). Then an instance of CSP(A2), say T(y), S(x, y) can be re-written as an instance of CSP(A1) R(y, z, y), R(x, y, w), R(y, y, x)

• 2. pp-interpretations (gadgets, possibly diff domains)
• 3. pp-constructions (gadgets + more)

Andrei Krokhin Constraints, Symmetry, and Complexity

Polymorphisms by example

Take any 2-Sat instance (x ∨ y) ∧ (y ∨ z) ∧ (y ∨ u) ∧ (x ∨ u) Take any three solutions a, b, c to this instance

Andrei Krokhin Constraints, Symmetry, and Complexity

Polymorphisms by example

Take any 2-Sat instance (x ∨ y) ∧ (y ∨ z) ∧ (y ∨ u) ∧ (x ∨ u) Take any three solutions a, b, c to this instance Apply the ternary majority operation m to a, b, c coordinate-wise (variables ordered here as x, y, z, u)

Andrei Krokhin Constraints, Symmetry, and Complexity

Polymorphisms by example

Take any 2-Sat instance (x ∨ y) ∧ (y ∨ z) ∧ (y ∨ u) ∧ (x ∨ u) Take any three solutions a, b, c to this instance Apply the ternary majority operation m to a, b, c coordinate-wise (variables ordered here as x, y, z, u) m m m m ↓ ↓ ↓ ↓ a = ( 1 1 1 ) sat b = ( 1 1 1 ) sat c = ( 1 ) sat m(a, b, c) = ( 1 1 ) sat

Andrei Krokhin Constraints, Symmetry, and Complexity

Polymorphisms

An operation f : Am → A is called a polymorphism of a k-ary relation R ⊆ Ak if for any k-tuples f f f ↓ ↓ ↓ ( a11 , . . . , a1k ) ∈ R . . . . . . . . . . . . ( am1 , . . . , amk ) ∈ R ⇓ ( f (a11, . . . , am1) , . . . , f (a1k, . . . , amk) ) ∈ R

Andrei Krokhin Constraints, Symmetry, and Complexity

Polymorphisms

An operation f : Am → A is called a polymorphism of a k-ary relation R ⊆ Ak if for any k-tuples f f f ↓ ↓ ↓ ( a11 , . . . , a1k ) ∈ R . . . . . . . . . . . . ( am1 , . . . , amk ) ∈ R ⇓ ( f (a11, . . . , am1) , . . . , f (a1k, . . . , amk) ) ∈ R Call f a polymorphism of A if it is such for all R in A. Notation: Pol(A).

Andrei Krokhin Constraints, Symmetry, and Complexity

More examples of polymorphisms

• 1. Let R = {r ∈ Zn

p | Ar = b} be an affine subspace of Zn p.

Andrei Krokhin Constraints, Symmetry, and Complexity

More examples of polymorphisms

• 1. Let R = {r ∈ Zn

p | Ar = b} be an affine subspace of Zn p.

Then f (x1, x2, x3) = x1 − x2 + x3 is a polymorphism of R:

Andrei Krokhin Constraints, Symmetry, and Complexity

More examples of polymorphisms

• 1. Let R = {r ∈ Zn

p | Ar = b} be an affine subspace of Zn p.

Then f (x1, x2, x3) = x1 − x2 + x3 is a polymorphism of R: If r1, r2, r3 ∈ R, i.e. Ar1 = b, Ar2 = b, Ar3 = b, then

Andrei Krokhin Constraints, Symmetry, and Complexity

More examples of polymorphisms

• 1. Let R = {r ∈ Zn

p | Ar = b} be an affine subspace of Zn p.

Then f (x1, x2, x3) = x1 − x2 + x3 is a polymorphism of R: If r1, r2, r3 ∈ R, i.e. Ar1 = b, Ar2 = b, Ar3 = b, then A(r1 − r2 + r3) = Ar1 − Ar2 + Ar3

Andrei Krokhin Constraints, Symmetry, and Complexity

More examples of polymorphisms

• 1. Let R = {r ∈ Zn

p | Ar = b} be an affine subspace of Zn p.

Then f (x1, x2, x3) = x1 − x2 + x3 is a polymorphism of R: If r1, r2, r3 ∈ R, i.e. Ar1 = b, Ar2 = b, Ar3 = b, then A(r1 − r2 + r3) = Ar1 − Ar2 + Ar3 = b − b + b = b.

Andrei Krokhin Constraints, Symmetry, and Complexity

More examples of polymorphisms

• 1. Let R = {r ∈ Zn

p | Ar = b} be an affine subspace of Zn p.

Then f (x1, x2, x3) = x1 − x2 + x3 is a polymorphism of R: If r1, r2, r3 ∈ R, i.e. Ar1 = b, Ar2 = b, Ar3 = b, then A(r1 − r2 + r3) = Ar1 − Ar2 + Ar3 = b − b + b = b.

• 2. If A = (A, E) is a digraph then f is a polymorphism of A if

(a1, b1), . . . , (an, bn) ∈ E ⇒ (f (a1, . . . , an), f (b1, . . . , bn)) ∈ E.

Andrei Krokhin Constraints, Symmetry, and Complexity

More examples of polymorphisms

• 1. Let R = {r ∈ Zn

p | Ar = b} be an affine subspace of Zn p.

Then f (x1, x2, x3) = x1 − x2 + x3 is a polymorphism of R: If r1, r2, r3 ∈ R, i.e. Ar1 = b, Ar2 = b, Ar3 = b, then A(r1 − r2 + r3) = Ar1 − Ar2 + Ar3 = b − b + b = b.

• 2. If A = (A, E) is a digraph then f is a polymorphism of A if

(a1, b1), . . . , (an, bn) ∈ E ⇒ (f (a1, . . . , an), f (b1, . . . , bn)) ∈ E.

• 3. Every polymorphism of 3-SAT is a projection (aka dictator),

i.e. f (x1, . . . , xn) = xi for some i.

Andrei Krokhin Constraints, Symmetry, and Complexity

More examples of polymorphisms

• 1. Let R = {r ∈ Zn

p | Ar = b} be an affine subspace of Zn p.

Then f (x1, x2, x3) = x1 − x2 + x3 is a polymorphism of R: If r1, r2, r3 ∈ R, i.e. Ar1 = b, Ar2 = b, Ar3 = b, then A(r1 − r2 + r3) = Ar1 − Ar2 + Ar3 = b − b + b = b.

• 2. If A = (A, E) is a digraph then f is a polymorphism of A if

(a1, b1), . . . , (an, bn) ∈ E ⇒ (f (a1, . . . , an), f (b1, . . . , bn)) ∈ E.

• 3. Every polymorphism of 3-SAT is a projection (aka dictator),

i.e. f (x1, . . . , xn) = xi for some i.

• 4. Every polymorphism of 3-Col is of the form

f (x1, . . . , xn) = π(xi) for some i ≤ n and permutation π

Andrei Krokhin Constraints, Symmetry, and Complexity

Modern proof of Schaefer’s classification for Boolean CSPs

Problem Polymorphism Complexity 0-valid constant 0 Easy 1-valid constant 1 Easy 2-Sat majority Easy Horn-Sat min Easy Dual H-Sat max Easy 3-Lin2 x − y + z Easy 3-Sat

• nly projections

Hard

Andrei Krokhin Constraints, Symmetry, and Complexity

Polymorphisms as symmetries

Objects capturing the symmetry of CSP(A): Aut(A) = {f : A → A automorph} automorphism group End(A) = {f : A → A homomorph} endomorphism monoid Pol(A) = {f : An → A homomorph} polymorphism clone

Andrei Krokhin Constraints, Symmetry, and Complexity

Polymorphisms as symmetries

Objects capturing the symmetry of CSP(A): Aut(A) = {f : A → A automorph} automorphism group End(A) = {f : A → A homomorph} endomorphism monoid Pol(A) = {f : An → A homomorph} polymorphism clone Clone = set of multivariate functions on A which

• 1. is closed under composition, and
• 2. contains all projections/dictators

Andrei Krokhin Constraints, Symmetry, and Complexity

Polymorphisms as symmetries

Objects capturing the symmetry of CSP(A): Aut(A) = {f : A → A automorph} automorphism group End(A) = {f : A → A homomorph} endomorphism monoid Pol(A) = {f : An → A homomorph} polymorphism clone Clone = set of multivariate functions on A which

• 1. is closed under composition, and
• 2. contains all projections/dictators

Example: trivial clone T , consisting of all projections.

Andrei Krokhin Constraints, Symmetry, and Complexity

Polymorphisms as symmetries

Objects capturing the symmetry of CSP(A): Aut(A) = {f : A → A automorph} automorphism group End(A) = {f : A → A homomorph} endomorphism monoid Pol(A) = {f : An → A homomorph} polymorphism clone Clone = set of multivariate functions on A which

• 1. is closed under composition, and
• 2. contains all projections/dictators

Example: trivial clone T , consisting of all projections. Aut(A), End(A) - no info about the complexity of CSP(A). Pol(A) - a lot of info.

Andrei Krokhin Constraints, Symmetry, and Complexity

Simulation vs. polymorphisms

Theorem (Birkhoff’35; Geiger’68; Bodnarchuk et al.’69; Bodirsky; Willard; Barto, Oprˇ sal,Pinsker’18)

Andrei Krokhin Constraints, Symmetry, and Complexity

Simulation vs. polymorphisms

Theorem (Birkhoff’35; Geiger’68; Bodnarchuk et al.’69; Bodirsky; Willard; Barto, Oprˇ sal,Pinsker’18) A pp-defines B iff Pol(A) ⊆ Pol(B).

Andrei Krokhin Constraints, Symmetry, and Complexity

Simulation vs. polymorphisms

Theorem (Birkhoff’35; Geiger’68; Bodnarchuk et al.’69; Bodirsky; Willard; Barto, Oprˇ sal,Pinsker’18) A pp-defines B iff Pol(A) ⊆ Pol(B). A pp-interprets B iff Pol(A) → Pol(B) (homomorphism).

Andrei Krokhin Constraints, Symmetry, and Complexity

Simulation vs. polymorphisms

Theorem (Birkhoff’35; Geiger’68; Bodnarchuk et al.’69; Bodirsky; Willard; Barto, Oprˇ sal,Pinsker’18) A pp-defines B iff Pol(A) ⊆ Pol(B). A pp-interprets B iff Pol(A) → Pol(B) (homomorphism). A pp-constructs B iff Pol(A) Pol(B) (height-1 homo).

Andrei Krokhin Constraints, Symmetry, and Complexity

Simulation vs. polymorphisms

Theorem (Birkhoff’35; Geiger’68; Bodnarchuk et al.’69; Bodirsky; Willard; Barto, Oprˇ sal,Pinsker’18) A pp-defines B iff Pol(A) ⊆ Pol(B). A pp-interprets B iff Pol(A) → Pol(B) (homomorphism). A pp-constructs B iff Pol(A) Pol(B) (height-1 homo). Remarks: Proof constructive ⇒ generic reduction CSP(B) CSP(A)

Andrei Krokhin Constraints, Symmetry, and Complexity

Simulation vs. polymorphisms

Theorem (Birkhoff’35; Geiger’68; Bodnarchuk et al.’69; Bodirsky; Willard; Barto, Oprˇ sal,Pinsker’18) A pp-defines B iff Pol(A) ⊆ Pol(B). A pp-interprets B iff Pol(A) → Pol(B) (homomorphism). A pp-constructs B iff Pol(A) Pol(B) (height-1 homo). Remarks: Proof constructive ⇒ generic reduction CSP(B) CSP(A) ξ : Pol(A) → Pol(B) iff it “preserves equations/identities”

— This allows applications of deep structural universal algebra

Andrei Krokhin Constraints, Symmetry, and Complexity

Simulation vs. polymorphisms

Theorem (Birkhoff’35; Geiger’68; Bodnarchuk et al.’69; Bodirsky; Willard; Barto, Oprˇ sal,Pinsker’18) A pp-defines B iff Pol(A) ⊆ Pol(B). A pp-interprets B iff Pol(A) → Pol(B) (homomorphism). A pp-constructs B iff Pol(A) Pol(B) (height-1 homo). Remarks: Proof constructive ⇒ generic reduction CSP(B) CSP(A) ξ : Pol(A) → Pol(B) iff it “preserves equations/identities”

— This allows applications of deep structural universal algebra

ξ : Pol(A) Pol(B) iff it “preserves ... of height 1”

— Not used in resolving Dichotomy Conj, but very important

Andrei Krokhin Constraints, Symmetry, and Complexity

Equations/identities

Identities = functional equations

Andrei Krokhin Constraints, Symmetry, and Complexity

Equations/identities

Identities = functional equations Ex.: A (ternary) majority operation is one satisfying identities m(x, x, y) = m(x, y, x) = m(y, x, x) = x.

Andrei Krokhin Constraints, Symmetry, and Complexity

Equations/identities

Identities = functional equations Ex.: A (ternary) majority operation is one satisfying identities m(x, x, y) = m(x, y, x) = m(y, x, x) = x. A random identity: g(f (x, y), z, x)) = g(x, y, y))

Andrei Krokhin Constraints, Symmetry, and Complexity

Equations/identities

Identities = functional equations Ex.: A (ternary) majority operation is one satisfying identities m(x, x, y) = m(x, y, x) = m(y, x, x) = x. A random identity: g(f (x, y), z, x)) = g(x, y, y)) Height 1 identity: exactly 1 function symbol on both sides, e.g. f (x, y) = f (y, x), but not f (x, f (y, z)) = f (f (x, y), z)

Andrei Krokhin Constraints, Symmetry, and Complexity

Equations/identities

Identities = functional equations Ex.: A (ternary) majority operation is one satisfying identities m(x, x, y) = m(x, y, x) = m(y, x, x) = x. A random identity: g(f (x, y), z, x)) = g(x, y, y)) Height 1 identity: exactly 1 function symbol on both sides, e.g. f (x, y) = f (y, x), but not f (x, f (y, z)) = f (f (x, y), z) Can speak about an identity being satisfied in Pol(A)

Andrei Krokhin Constraints, Symmetry, and Complexity

Equations/identities

Identities = functional equations Ex.: A (ternary) majority operation is one satisfying identities m(x, x, y) = m(x, y, x) = m(y, x, x) = x. A random identity: g(f (x, y), z, x)) = g(x, y, y)) Height 1 identity: exactly 1 function symbol on both sides, e.g. f (x, y) = f (y, x), but not f (x, f (y, z)) = f (f (x, y), z) Can speak about an identity being satisfied in Pol(A) Can consider systems of identities

Andrei Krokhin Constraints, Symmetry, and Complexity

Equations/identities

Identities = functional equations Ex.: A (ternary) majority operation is one satisfying identities m(x, x, y) = m(x, y, x) = m(y, x, x) = x. A random identity: g(f (x, y), z, x)) = g(x, y, y)) Height 1 identity: exactly 1 function symbol on both sides, e.g. f (x, y) = f (y, x), but not f (x, f (y, z)) = f (f (x, y), z) Can speak about an identity being satisfied in Pol(A) Can consider systems of identities Compare

— f (x1, x2, x3) = f (x1, x3, x2) trivial symmetry — f (x1, x2, x3) = f (x2, x3, x1) non-trivial symmetry

Andrei Krokhin Constraints, Symmetry, and Complexity

Preserving identities

Pol(A) → Pol(B) means

— “each system of identities satisfied in Pol(A) is also satisfied in Pol(B)”, or

Andrei Krokhin Constraints, Symmetry, and Complexity

Preserving identities

Pol(A) → Pol(B) means

— “each system of identities satisfied in Pol(A) is also satisfied in Pol(B)”, or — “B has more symmetries than A” - in a qualitative way

Andrei Krokhin Constraints, Symmetry, and Complexity

Preserving identities

Pol(A) → Pol(B) means

— “each system of identities satisfied in Pol(A) is also satisfied in Pol(B)”, or — “B has more symmetries than A” - in a qualitative way

Ex.: If Pol(A) has a commutative (or associate, or majority)

• peration then so does Pol(B)

Andrei Krokhin Constraints, Symmetry, and Complexity

Preserving identities

Pol(A) → Pol(B) means

— “each system of identities satisfied in Pol(A) is also satisfied in Pol(B)”, or — “B has more symmetries than A” - in a qualitative way

Ex.: If Pol(A) has a commutative (or associate, or majority)

• peration then so does Pol(B)

Systems of (h1) identites sat in Pol(A) = measure of symmetry.

Andrei Krokhin Constraints, Symmetry, and Complexity

Tractability conjecture

Theorem If Pol(A) → T then CSP(A) is NP-complete.

Andrei Krokhin Constraints, Symmetry, and Complexity

Tractability conjecture

Theorem If Pol(A) → T then CSP(A) is NP-complete.

• Proof. If B is 3-SAT then Pol(B) = T , so A can simulate

(pp-interpret) 3-SAT, and hence 3-SAT reduces to CSP(A).

Andrei Krokhin Constraints, Symmetry, and Complexity

Tractability conjecture

Theorem If Pol(A) → T then CSP(A) is NP-complete.

• Proof. If B is 3-SAT then Pol(B) = T , so A can simulate

(pp-interpret) 3-SAT, and hence 3-SAT reduces to CSP(A). Note: Identities satisfied in T are trivial - satisfied in each Pol(A). If Pol(A) → T then Pol(A) satisfies only trivial identities.

Andrei Krokhin Constraints, Symmetry, and Complexity

Tractability conjecture

Theorem If Pol(A) → T then CSP(A) is NP-complete.

• Proof. If B is 3-SAT then Pol(B) = T , so A can simulate

(pp-interpret) 3-SAT, and hence 3-SAT reduces to CSP(A). Note: Identities satisfied in T are trivial - satisfied in each Pol(A). If Pol(A) → T then Pol(A) satisfies only trivial identities. Conjecture (CSP Tractability Conjecture; Bulatov, Jeavons, AK’05; many others) If Pol(A) → T then CSP(A) is in P.

Andrei Krokhin Constraints, Symmetry, and Complexity

Tractability conjecture

Theorem If Pol(A) → T then CSP(A) is NP-complete.

• Proof. If B is 3-SAT then Pol(B) = T , so A can simulate

(pp-interpret) 3-SAT, and hence 3-SAT reduces to CSP(A). Note: Identities satisfied in T are trivial - satisfied in each Pol(A). If Pol(A) → T then Pol(A) satisfies only trivial identities. Conjecture (CSP Tractability Conjecture; Bulatov, Jeavons, AK’05; many others) If Pol(A) → T then CSP(A) is in P. If Pol(A) → T , Pol(A) satisfies some non-trivial identities.

Andrei Krokhin Constraints, Symmetry, and Complexity

Algebraic dichotomy (picture) Relational structures

3-Sat

Polymorphisms satisfying non-trivial identities

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Non-trivial symmetries ⇒ strong symmetries

Theorem TFAE:

• 1. Pol(A) → T , i.e. Pol(A) satisfies some non-trivial identities.

Andrei Krokhin Constraints, Symmetry, and Complexity

Non-trivial symmetries ⇒ strong symmetries

Theorem TFAE:

• 1. Pol(A) → T , i.e. Pol(A) satisfies some non-trivial identities.
• 2. A has a weak near-unanimity polym’m [M´

aroti,McKenzie’08] f (y, x, . . . , x, x) = f (x, y, . . . , x, x) = . . . = f (x, x, . . . , x, y)

Andrei Krokhin Constraints, Symmetry, and Complexity

Non-trivial symmetries ⇒ strong symmetries

Theorem TFAE:

• 1. Pol(A) → T , i.e. Pol(A) satisfies some non-trivial identities.
• 2. A has a weak near-unanimity polym’m [M´

aroti,McKenzie’08] f (y, x, . . . , x, x) = f (x, y, . . . , x, x) = . . . = f (x, x, . . . , x, y)

• 3. A has a cyclic polymorphism [Barto,Kozik’12]

f (x1, x2, x3, . . . , xn) = f (x2, x3, . . . , xn, x1)

Andrei Krokhin Constraints, Symmetry, and Complexity

Non-trivial symmetries ⇒ strong symmetries

Theorem TFAE:

• 1. Pol(A) → T , i.e. Pol(A) satisfies some non-trivial identities.
• 2. A has a weak near-unanimity polym’m [M´

aroti,McKenzie’08] f (y, x, . . . , x, x) = f (x, y, . . . , x, x) = . . . = f (x, x, . . . , x, y)

• 3. A has a cyclic polymorphism [Barto,Kozik’12]

f (x1, x2, x3, . . . , xn) = f (x2, x3, . . . , xn, x1)

• 4. A has a Siggers polymorphism [Siggers’09,KMM’14]

f (r, a, r, e) = f (a, r, e, a)

Andrei Krokhin Constraints, Symmetry, and Complexity

Efficient Algorithms for CSPs

Constraint propagation: given a CSP instance, repeatedly derive new constraints

Andrei Krokhin Constraints, Symmetry, and Complexity

Efficient Algorithms for CSPs

Constraint propagation: given a CSP instance, repeatedly derive new constraints If want to work in poly-time, need to

Andrei Krokhin Constraints, Symmetry, and Complexity

Efficient Algorithms for CSPs

Constraint propagation: given a CSP instance, repeatedly derive new constraints If want to work in poly-time, need to

• 1. bound the size of subinstances used for derivation,

Andrei Krokhin Constraints, Symmetry, and Complexity

Efficient Algorithms for CSPs

Constraint propagation: given a CSP instance, repeatedly derive new constraints If want to work in poly-time, need to

• 1. bound the size of subinstances used for derivation,
• 2. or use a compact representation for derived constraints

The power of each approach is well understood.

Andrei Krokhin Constraints, Symmetry, and Complexity

Efficient Algorithms for CSPs

Constraint propagation: given a CSP instance, repeatedly derive new constraints If want to work in poly-time, need to

• 1. bound the size of subinstances used for derivation,
• 2. or use a compact representation for derived constraints

The power of each approach is well understood. How polymorphisms are used: combine (partial) solutions to produce a better one

Andrei Krokhin Constraints, Symmetry, and Complexity

Efficient Algorithms for CSPs

Constraint propagation: given a CSP instance, repeatedly derive new constraints If want to work in poly-time, need to

• 1. bound the size of subinstances used for derivation,
• 2. or use a compact representation for derived constraints

The power of each approach is well understood. How polymorphisms are used: combine (partial) solutions to produce a better one directly - algorithm actually combines solutions (Approach 2) indirectly - to prove correctness (Approach 1)

Andrei Krokhin Constraints, Symmetry, and Complexity

Efficient Algorithms for CSPs

Constraint propagation: given a CSP instance, repeatedly derive new constraints If want to work in poly-time, need to

• 1. bound the size of subinstances used for derivation,
• 2. or use a compact representation for derived constraints

The power of each approach is well understood. How polymorphisms are used: combine (partial) solutions to produce a better one directly - algorithm actually combines solutions (Approach 2) indirectly - to prove correctness (Approach 1) identities say which algorithm can be used

Andrei Krokhin Constraints, Symmetry, and Complexity

The “few subpowers” algorithm - Approach 2

Can solve a system of linear equations over Zp as follows:

Andrei Krokhin Constraints, Symmetry, and Complexity

The “few subpowers” algorithm - Approach 2

Can solve a system of linear equations over Zp as follows:

• 1. Let Bi be the basis for solution set of the first i equations

Andrei Krokhin Constraints, Symmetry, and Complexity

The “few subpowers” algorithm - Approach 2

Can solve a system of linear equations over Zp as follows:

• 1. Let Bi be the basis for solution set of the first i equations
• 2. Can find Bi from Bi−1 and the i-th equation

Andrei Krokhin Constraints, Symmetry, and Complexity

The “few subpowers” algorithm - Approach 2

Can solve a system of linear equations over Zp as follows:

• 1. Let Bi be the basis for solution set of the first i equations
• 2. Can find Bi from Bi−1 and the i-th equation
• 3. Size of Bi is bounded by n (the number of variabless)
• 4. At the end, have a compact representation for all solutions

Andrei Krokhin Constraints, Symmetry, and Complexity

The “few subpowers” algorithm - Approach 2

Can solve a system of linear equations over Zp as follows:

• 1. Let Bi be the basis for solution set of the first i equations
• 2. Can find Bi from Bi−1 and the i-th equation
• 3. Size of Bi is bounded by n (the number of variabless)
• 4. At the end, have a compact representation for all solutions

The above can be extended to work with general constraints

Andrei Krokhin Constraints, Symmetry, and Complexity

The “few subpowers” algorithm - Approach 2

Can solve a system of linear equations over Zp as follows:

• 1. Let Bi be the basis for solution set of the first i equations
• 2. Can find Bi from Bi−1 and the i-th equation
• 3. Size of Bi is bounded by n (the number of variabless)
• 4. At the end, have a compact representation for all solutions

The above can be extended to work with general constraints (a) Bi can be a “generating set”, need it to be small (O(nc)) (b) need to be able perform the above item (2) efficiently

Andrei Krokhin Constraints, Symmetry, and Complexity

The “few subpowers” algorithm - Approach 2

Can solve a system of linear equations over Zp as follows:

• 1. Let Bi be the basis for solution set of the first i equations
• 2. Can find Bi from Bi−1 and the i-th equation
• 3. Size of Bi is bounded by n (the number of variabless)
• 4. At the end, have a compact representation for all solutions

The above can be extended to work with general constraints (a) Bi can be a “generating set”, need it to be small (O(nc)) (b) need to be able perform the above item (2) efficiently Theorem (Idziak, Markovic, McKenzie, Valeriote, Willard’10) Certain identities sat in Pol(A) ⇔ (a). If have (a), can do (b).

Andrei Krokhin Constraints, Symmetry, and Complexity

Local consistency algorithm - Approach 1

Andrei Krokhin Constraints, Symmetry, and Complexity

Local consistency algorithm - Approach 1

Roughly: Perform local propagation until stable. If not refuted, return “sat”.

Andrei Krokhin Constraints, Symmetry, and Complexity

Local consistency algorithm - Approach 1

Roughly: Perform local propagation until stable. If not refuted, return “sat”. (Standard algorithm for 2-Sat and Horn 3-Sat).

Andrei Krokhin Constraints, Symmetry, and Complexity

Local consistency algorithm - Approach 1

Roughly: Perform local propagation until stable. If not refuted, return “sat”. (Standard algorithm for 2-Sat and Horn 3-Sat). More precisely: Fix integers k ≤ ℓ. (k, ℓ)-algorithm: Repeatedly derive the strongest constraints

• n k vars by considering ℓ vars at a time.

Andrei Krokhin Constraints, Symmetry, and Complexity

Local consistency algorithm - Approach 1

Roughly: Perform local propagation until stable. If not refuted, return “sat”. (Standard algorithm for 2-Sat and Horn 3-Sat). More precisely: Fix integers k ≤ ℓ. (k, ℓ)-algorithm: Repeatedly derive the strongest constraints

• n k vars by considering ℓ vars at a time.

If the instance is refuted, output “no”. Otherwise, “yes”. “No” answers are always correct.

Andrei Krokhin Constraints, Symmetry, and Complexity

Local consistency algorithm - Approach 1

Roughly: Perform local propagation until stable. If not refuted, return “sat”. (Standard algorithm for 2-Sat and Horn 3-Sat). More precisely: Fix integers k ≤ ℓ. (k, ℓ)-algorithm: Repeatedly derive the strongest constraints

• n k vars by considering ℓ vars at a time.

If the instance is refuted, output “no”. Otherwise, “yes”. “No” answers are always correct. If “yes” answers are also correct for each instance of CSP(A), we say that A has width (k, ℓ). if some (k, ℓ) work for A, say A has bounded width.

Andrei Krokhin Constraints, Symmetry, and Complexity

Local consistency algorithm - Approach 1

Roughly: Perform local propagation until stable. If not refuted, return “sat”. (Standard algorithm for 2-Sat and Horn 3-Sat). More precisely: Fix integers k ≤ ℓ. (k, ℓ)-algorithm: Repeatedly derive the strongest constraints

• n k vars by considering ℓ vars at a time.

If the instance is refuted, output “no”. Otherwise, “yes”. “No” answers are always correct. If “yes” answers are also correct for each instance of CSP(A), we say that A has width (k, ℓ). if some (k, ℓ) work for A, say A has bounded width. Equivalent notions: treewidth duality, Datalog, pebble games, etc

Andrei Krokhin Constraints, Symmetry, and Complexity

Picture for bounded width?

3-Lin2 3-Lin3 3-Linp . . . . . . . . .

Strong enough polymorphisms (symmetries)

Andrei Krokhin Constraints, Symmetry, and Complexity

Bounded width

Theorem TFAE:

• 1. A doesn’t pp-construct 3-Linp for any p;

Andrei Krokhin Constraints, Symmetry, and Complexity

Bounded width

Theorem TFAE:

• 1. A doesn’t pp-construct 3-Linp for any p;
• 2. A has polymorphisms f3, f4 such that

[Kozik et al.’14] f3(x, x, y) = f3(x, y, x) = f3(y, x, x) = f4(x, x, x, y) = . . . = f4(y, x, x, x)

Andrei Krokhin Constraints, Symmetry, and Complexity

Bounded width

Theorem TFAE:

• 1. A doesn’t pp-construct 3-Linp for any p;
• 2. A has polymorphisms f3, f4 such that

[Kozik et al.’14] f3(x, x, y) = f3(x, y, x) = f3(y, x, x) = f4(x, x, x, y) = . . . = f4(y, x, x, x)

• 3. A has bounded width

[Barto, Kozik’09-14]

• 4. A has width (2,3)

[Barto’16, Bulatov]

Andrei Krokhin Constraints, Symmetry, and Complexity

Bounded width

Theorem TFAE:

• 1. A doesn’t pp-construct 3-Linp for any p;
• 2. A has polymorphisms f3, f4 such that

[Kozik et al.’14] f3(x, x, y) = f3(x, y, x) = f3(y, x, x) = f4(x, x, x, y) = . . . = f4(y, x, x, x)

• 3. A has bounded width

[Barto, Kozik’09-14]

• 4. A has width (2,3)

[Barto’16, Bulatov]

• 5. CSP(A) is robustly solvable

[Barto, Kozik’12-16]

— For each almost satisfiable instance, an almost satisfying assignment can be found in poly-time.

Andrei Krokhin Constraints, Symmetry, and Complexity

Bounded width

Theorem TFAE:

• 1. A doesn’t pp-construct 3-Linp for any p;
• 2. A has polymorphisms f3, f4 such that

[Kozik et al.’14] f3(x, x, y) = f3(x, y, x) = f3(y, x, x) = f4(x, x, x, y) = . . . = f4(y, x, x, x)

• 3. A has bounded width

[Barto, Kozik’09-14]

• 4. A has width (2,3)

[Barto’16, Bulatov]

• 5. CSP(A) is robustly solvable

[Barto, Kozik’12-16]

— For each almost satisfiable instance, an almost satisfying assignment can be found in poly-time.

Polymorphisms are used only to prove correctness of the algorithm.

Andrei Krokhin Constraints, Symmetry, and Complexity

Bulatov’s and Zhuk’s algorithms (very roughly), 1

Theorem (Bulatov’17, Zhuk’17) If A has a weak near-unanimity polymorphism (equivalently, Pol(A) → T ) then CSP(A) is in P. On a VERY high level, the two algorithms are similar:

Andrei Krokhin Constraints, Symmetry, and Complexity

Bulatov’s and Zhuk’s algorithms (very roughly), 1

Theorem (Bulatov’17, Zhuk’17) If A has a weak near-unanimity polymorphism (equivalently, Pol(A) → T ) then CSP(A) is in P. On a VERY high level, the two algorithms are similar: Both heavily use local propagation.

Andrei Krokhin Constraints, Symmetry, and Complexity

Bulatov’s and Zhuk’s algorithms (very roughly), 1

Theorem (Bulatov’17, Zhuk’17) If A has a weak near-unanimity polymorphism (equivalently, Pol(A) → T ) then CSP(A) is in P. On a VERY high level, the two algorithms are similar: Both heavily use local propagation. Both are recursive: initially, let dom(v) = A for each variable v if some dom(v) can be reduced without losing all solutions, reduce it and recurse

Andrei Krokhin Constraints, Symmetry, and Complexity

Bulatov’s and Zhuk’s algorithms (very roughly), 1

Theorem (Bulatov’17, Zhuk’17) If A has a weak near-unanimity polymorphism (equivalently, Pol(A) → T ) then CSP(A) is in P. On a VERY high level, the two algorithms are similar: Both heavily use local propagation. Both are recursive: initially, let dom(v) = A for each variable v if some dom(v) can be reduced without losing all solutions, reduce it and recurse if an instance can be split/decomposed into a constant number of disjoint instances, split/decompose and recurse this can be done in many situations

Andrei Krokhin Constraints, Symmetry, and Complexity

Bulatov’s and Zhuk’s algorithms (very roughly), 2

The base cases for recursion (roughly) are: (B and Z) all domains 1-element → solution (B and Z) some domain empty → no solution

Andrei Krokhin Constraints, Symmetry, and Complexity

Bulatov’s and Zhuk’s algorithms (very roughly), 2

The base cases for recursion (roughly) are: (B and Z) all domains 1-element → solution (B and Z) some domain empty → no solution (B) few subpowers algorithm is applicable (Z) systems of linear equations over Zp

Andrei Krokhin Constraints, Symmetry, and Complexity

Bulatov’s and Zhuk’s algorithms (very roughly), 2

The base cases for recursion (roughly) are: (B and Z) all domains 1-element → solution (B and Z) some domain empty → no solution (B) few subpowers algorithm is applicable (Z) systems of linear equations over Zp However, the recursion is organised very differently: (B) heavily uses deep algebraic structure (Z) still very algebraic, but more elementary (B and Z) a whole range of diff algebraic tricks

Andrei Krokhin Constraints, Symmetry, and Complexity

Feder-Vardi conjecture proved – End of the road ???

Andrei Krokhin Constraints, Symmetry, and Complexity

Feder-Vardi conjecture proved – End of the road ???

Actually, we are rather closer to the beginning!

Andrei Krokhin Constraints, Symmetry, and Complexity

Message

• 1. Decision CSP: Can all constraints be satisfied?
• 2. Counting CSP: Count the number of solutions
• 3. Max CSP: Find a map satisfying max number of constraints
• 4. Approx Max CSP: Satisfy c × Opt number of constraints
• 5. Approx Min CSP: assuming 1 − ǫ fraction of constraints can

be satisfied, find a map satisfying ≥ 1 − g(ǫ) fraction.

• 6. Promise CSP: ... tomorrow

Each of the above has an appropriate notion of symmetry!

lack of symmetries ⇒ hardness symmetries ⇒ efficient algorithms symmetries of higher dimension/arity are important

Andrei Krokhin Constraints, Symmetry, and Complexity

Useful surveys

Much of what is covered in this lecture can be found in survey Polymorphisms, and how to use them.

• L. Barto, A. Krokhin, and R. Willard.

It is written specifically for those without algebra background! See also the full (open-access) volume of surveys: The Constraint Satisfaction Problem: Complexity and

• Approximability. Editors: A. Krokhin and S. ˇ

Zivn´ y. Dagstuhl Follow-Ups series, Volume 7, 2017. http://drops.dagstuhl.de/portals/dfu/index.php?semnr=16027

Andrei Krokhin Constraints, Symmetry, and Complexity