Constraints, Symmetry, and Complexity · Andrei Krokhin Constraints, Symmetry, and Complexity....

Constraints, Symmetry, and Complexity

Andrei Krokhin

Andrei Krokhin Constraints, Symmetry, and Complexity

Message

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

Message

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)

Message

Main Achievement: better understanding of commonstructure 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

Message

Constraint Satisfaction Problem (CSP)

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

Definition

Examples and a conjecture

k-Col: A = ([k]; {6=})

k-NAE: A = ([k]; {(a, b, c) ∈ [k]3 | a 6= b ∨ a 6= c ∨ b 6= 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.

k-Col: A = ([k]; {6=})k-NAE: A = ([k]; {(a, b, c) ∈ [k]3 | a 6= b ∨ a 6= c ∨ b 6= c})

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

3-Sat: A = ({0, 1}; (x ∨ y ∨ z), (x ∨ y ∨ ¬z), . . .)

3-Linp: A = (Zp; x + y + z = 0, x + 2y + 3z = 7, . . .)

3-Sat: A = ({0, 1}; (x ∨ y ∨ z), (x ∨ y ∨ ¬z), . . .)

3-Linp: A = (Zp; x + y + z = 0, x + 2y + 3z = 7, . . .)

3-Sat: A = ({0, 1}; (x ∨ y ∨ z), (x ∨ y ∨ ¬z), . . .)

3-Linp: A = (Zp; x + y + z = 0, x + 2y + 3z = 7, . . .)

3-Sat: A = ({0, 1}; (x ∨ y ∨ z), (x ∨ y ∨ ¬z), . . .)

3-Linp: A = (Zp; x + y + z = 0, x + 2y + 3z = 7, . . .)

3-Sat: A = ({0, 1}; (x ∨ y ∨ z), (x ∨ y ∨ ¬z), . . .)

3-Linp: A = (Zp; x + y + z = 0, x + 2y + 3z = 7, . . .)

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, Zivny’16], [Kolmogorov, AK, Rolınek’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 constraintscan 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

Algebraic approach to CSP (very high-level view)

Relationalstructures

“can simulate”

. . . . . . . . .

. . . . . .

“can simulate”

. . . . . . . . .

. . . . . .

Strong enoughpolymorphisms(symmetries)

“can simulate”3-Sat

. . . . . . . . .

. . . . . .

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)

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

Ex.: Let

A1 = (A;R), R ternary, and A2 = (A;T ,S) be s.t.

T (y),S(x , y)

R(y , z , y),R(x , y ,w),R(y , y , x)

A1 = (A;R), R ternary, and A2 = (A;T , S) be s.t.

T (y),S(x , y)

R(y , z , y),R(x , y ,w),R(y , y , x)

T (y),S(x , y)

R(y , z , y),R(x , y ,w),R(y , y , x)

T (y),S(x , y)

R(y , z , y),R(x , y ,w),R(y , y , x)

T (y),S(x , y)

R(y , z , y),R(x , y ,w),R(y , y , x)

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

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 0 ) sat

b = ( 1 1 0 1 ) sat

c = ( 1 0 0 0 ) sat

m(a,b, c) = ( 1 1 0 0 ) sat

m m m m

↓ ↓ ↓ ↓a = ( 1 1 1 0 ) sat

b = ( 1 1 0 1 ) sat

c = ( 1 0 0 0 ) sat

m(a,b, c) = ( 1 1 0 0 ) sat

m m m m

↓ ↓ ↓ ↓a = ( 1 1 1 0 ) sat

b = ( 1 1 0 1 ) sat

c = ( 1 0 0 0 ) sat

m(a,b, c) = ( 1 1 0 0 ) sat

Polymorphisms

An operation f : Am → A is called a polymorphism of a k-aryrelation 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).

Polymorphisms

An operation f : Am → A is called a polymorphism of a k-aryrelation 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).

More examples of polymorphisms

1. Let R = {r ∈ Znp | Ar = b} be an affine subspace of Zn

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

A(r1 − r2 + r3) = Ar1 − Ar2 + Ar3 = b− b + b = b.

A(r1 − r2 + r3) = Ar1 − Ar2 + Ar3

= b− b + b = b.

A(r1 − r2 + r3) = Ar1 − Ar2 + Ar3 = b− b + b = b.

Modern proof of Schaefer’s classification for Boolean CSPs

Problem Polymorphism Complexity

0-valid constant 0 Easy1-valid constant 1 Easy2-Sat majority Easy

Horn-Sat min EasyDual H-Sat max Easy

3-Lin2 x − y + z Easy3-Sat only projections Hard

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.

Simulation vs. polymorphisms

Theorem (Birkhoff’35; Geiger’68; Bodnarchuk et al.’69; Bodirsky;

Willard; Barto, Oprsal,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) 99K 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) 99K Pol(B) iff it “preserves ... of height 1”

— Not used in resolving Dichotomy Conj, but very important

Remarks:

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

m(x , x , y) = m(x , y , x) = m(y , x , x) = x .

Compare

m(x , x , y) = m(x , y , x) = m(y , x , x) = x .

Compare

m(x , x , y) = m(x , y , x) = m(y , x , x) = x .

Compare

m(x , x , y) = m(x , y , x) = m(y , x , x) = x .

Compare

m(x , x , y) = m(x , y , x) = m(y , x , x) = x .

Compare

m(x , x , y) = m(x , y , x) = m(y , x , x) = x .

Compare

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)

operation then so does Pol(B)

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

Pol(B)”, or

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).

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) 6→ T then CSP(A) is in P.

If Pol(A) 6→ T , Pol(A) satisfies some non-trivial identities.

Theorem

many others)

Theorem

many others)

Theorem

many others)

Theorem

many others)

Algebraic dichotomy (picture)

Polymorphisms satisfyingnon-trivial identities

Non-trivial symmetries ⇒ strong symmetries

Theorem

1. Pol(A) 6→ T , i.e. Pol(A) satisfies some non-trivial identities.

2. A has a weak near-unanimity polym’m [Maroti,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)

Theorem

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

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

Theorem

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

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

Theorem

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

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

Efficient Algorithms for CSPs

Constraint propagation: given a CSP instance, repeatedly derivenew 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

Constraint propagation: given a CSP instance, repeatedly derivenew constraintsIf want to work in poly-time, need to

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).

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 ≤ `.

(k , `)-algorithm: Repeatedly derive the strongest constraints

on k vars by considering ` 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, `).

if some (k, `) work for A, say A has bounded width.

Equivalent notions: treewidth duality, Datalog, pebble games, etc

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

(Standard algorithm for 2-Sat and Horn 3-Sat).

More precisely:

Picture for bounded width?

3-Lin2 3-Lin3 3-Linp

. . . . . . . . .

Bounded width

Theorem

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.

Bounded width

Theorem

f3(x , x , y) = f3(x , y , x) = f3(y , x , x) =

f4(x , x , x , y) = . . . = f4(y , x , x , x)

Bounded width

Theorem

f3(x , x , y) = f3(x , y , x) = f3(y , x , x) =

f4(x , x , x , y) = . . . = f4(y , x , x , x)

Bounded width

Theorem

f3(x , x , y) = f3(x , y , x) = f3(y , x , x) =

f4(x , x , x , y) = . . . = f4(y , x , x , x)

Bounded width

Theorem

f3(x , x , y) = f3(x , y , x) = f3(y , x , x) =

f4(x , x , x , y) = . . . = f4(y , x , x , x)

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) 6→ 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

On a VERY high level, the two algorithms are similar:Both heavily use local propagation.

Both are recursive:

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

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

Actually, we are rather closer to thebeginning!

Message

5. Approx Min CSP: assuming 1− ε fraction of constraints can

be satisfied, find a map satisfying ≥ 1− g(ε) 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

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. Zivny.

Dagstuhl Follow-Ups series, Volume 7, 2017.

http://drops.dagstuhl.de/portals/dfu/index.php?semnr=16027

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