Algebraic Approach to Promise Constraint Satisfaction Jakub Bul n - - PowerPoint PPT Presentation

Algebraic Approach to Promise Constraint Satisfaction

Jakub Bul´ ın

Charles University

June 10, 2019

Joint work with Libor Barto (Charles University), Andrei Krokhin&Jakub Oprˇ sal (Durham University)

Constraint Satisfaction

Many natural computational tasks. . .

• V – a finite set of variables,
• A – a finite set of values,
• C = {C1, . . . , Cm} – finitely many constraints Ci = (¯

si, Ri), where ¯ si is a ki-tuple of variables and Ri ⊆ Aki

• Q: Is there a solution, i.e. ϕ : V → A such that ϕ(¯

si) ∈ Ri? NP-complete, a natural restriction: fixed template CSPs

• Fix A and finitely many admissible relations Ri, i.e. a finite

relational structure A = (A; R1, . . . , Rk)

• Goal: characterize relational structures wrt. complexity

(and other algorithmic properties) of the corresponding CSP

The CSP dichotomy

CSP(A):

• Input: a finite structure X in the same language
• Decide: Is there a homomorphism X → A?

A decades-long research program. . .

• Conjecture For every A, CSP(A) is in P or NP-complete. [Feder,

Vardi ’93]

• Theorem True for Boolean templates. [Schaefer ’78]
• Theorem True for graphs. [Hell, Neˇ

setˇ ril ’90]

. . . Cross-fertilization with universal algebra. . .

• Theorem True. [Bulatov ’17; Zhuk ’17]

Polymorphisms in a nutshell

A polymorphism of CSP(A)

• a “high-dimensional symmetry” of solution spaces
• a function f : An → A preserving all the constraint relations,

i.e. for each RA and ai ∈ RA, f (a1, . . . , an) ∈ RA

• a multivariate endomorphism f : An → A

Key insight: “more symmetric is easier”

• S ⊆ Ak is the solution set to some instance ⇔ preserved by all

f ∈ Pol(A)

• adding S to A cannot change the complexity of CSP(A)
• Pol(A) ⊆ Pol(B) ⇒ CSP(B) ≤L CSP(A)

[Jeavons, Cohen, Gyssens ’97]

Abstract polymorphisms

• Pol(A) is a clone = contains projections (aka dictators), closed

under composition

• clone homomorphism: Φ : C1 → C2 preserving projections and

composition

• Pol(A) → Pol(B) ⇒ CSP(B) ≤L CSP(A)

“Birkhoff’s HSP Theorem” [Bulatov, Jeavons, Krokhin ’05]

• abstract clone: clones modulo homomorphic equivalence

= a set of equational (Мальцев) conditions

• algebraic dichotomy: CSP(A) is NP-complete

⇔ Pol(A) → P2 (projections on {0, 1}) ⇔ Pol(A) satisfies no nontrivial equational conditions ⇔ the abstract clone is trivial . . . and otherwise it is in P (that’s the hard part)

Promise constraint satisfaction

Fix a pair of finite structures A, B such that A → B

PCSP(A, B) [Brakensiek, Guruswami ’16]

• Input: a finite structure X in the same language
• Promise: either X → A or X → B
• ACCEPT if X → A, REJECT if X → B
• Related to approximation: given a satisfiable instance of a

hard problem, find an approx. solution (relaxed constraints)

• Generalizes CSP:
• CSP(A) = PCSP(A, A)
• empty promise, all inputs valid, no approximation

A famous example

Approximate graph coloring [Garey, Johnson ’76]

• Search: find a d-coloring of a c-colorable graph (c < d)
• Decide: distinguish between c-colorable graphs and graphs

that are not even d-colorable = PCSP(Kc, Kd) Believed NP-hard for all 3 ≤ c < d, known for

• (3, 4) [Khanna, Linial, Safra ’00]
• (K, 2Ω(K

1 3 )) for big enough K [Huang ’13]

• (4, 6), (k, 2k − 2) [Brakensiek, Guruswami ’16]
• (3, 5), (k, 2k − 1) [JB, Krokhin, Oprˇ

sal ’19]

Polymorphisms for PCSP

A polymorphism of PCSP(A, B)

• f : An → B s.t. a1, . . . , an ∈ RA ⇒ f (a1, . . . , an) ∈ RB
• a multivariate ✘✘

✘ ❳❳ ❳

endohomomorphism f : An → B

• Pol(A1, B1) ⊆ Pol(A2, B2) ⇒ PCSP(A2, B2) ≤L PCSP(A1, B1)

[Brakensiek, Guruswami ’18]

• Pol(A, B) is not a clone! (not closed under composition)
• but it is a minion = closed under identification minors, i.e.,

identify, permute, or add dummy variables: g(x1, . . . , xm) = f (xσ(1), . . . , xσ(m)), for any σ : [m] → [n]

Abstract minions

• minor homomorphism = Φ : M1 → M2 preserving minors
• minor condition = a finite set of minor identities:

g(x1, . . . , xm) ≈ f (xσ(1), . . . , xσ(m))

• minor homomorphism ⇔ preserves minor conditions
• abstract minion: minions modulo homomorphic equivalence

= a set of minor conditions

“Linear Birkhoff’s HSP Theorem” [JB, Krokhin, Oprˇ

sal ’19]

If Pol(A2, B2) satisfies all minor conditions satisfied by Pol(A1, B1), then PCSP(A2, B2) ≤L PCSP(A1, B1).

• for CSPs partly already in [Barto, Oprˇ

sal, Pinsker ’18]

• a completely new, straightforward proof

Sketch of proof

An intermediate problem: promise minor condition PMCM(N)

• Input: a minor condition Σ in at most N variables
• ACCEPT if Σ is trivial, REJECT if M |

= Σ

1 PCSP(A2, B2) ≤L PMCPol(A2,B2)(N) for N = maxR{|A2|, |RA2|}

• a construction: from X to Σ (on the next slide)
• X → A2 ⇒ Σ trivial
• Pol(A2, B2) |

= Σ ⇒ X → B2

2 PMCPol(A2,B2)(N) ≤L PMCPol(A1,B1)(N) for all N > 0

• trivially from the assumption

3 PMCPol(A1,B1)(N) ≤L PCSP(A1, B1) for all N > 0

• “indicator construction”: variables f (a1, . . . , an), assert that f ’s

are polymorphisms, add = where required by the identities

Example of construction

• Let A2 = {0, 1}; NAE
• Function symbols:
• fx of arity |A2| = 2 for x ∈ X,
• gC of arity |RA2| = 6 for every constraint C : (x, y, z) ∈ R
• Identities:for every constraint C : (x, y, z) ∈ R add

fx(u0, u1) ≈ gC(u0, u0, u1, u1, u1, u0) fy(u0, u1) ≈ gC(u0, u1, u0, u1, u0, u1) fz(u0, u1) ≈ gC(u1, u0, u0, u0, u1, u1)

• X → A2 ⇒ Σ trivial: fx := projection to ϕ(x)th coordinate
• Pol(A2, B2) |

= Σ ⇒ X → B2: ϕ(x) := fx(0, 1)

Applications

Hardness

A (super)natural reduction from gap label cover via the long code test (Appendix A)

• label cover instances = minor conditions
• uniform explanation (simplification, strengthening) of known

hardness proofs, e.g. approximate graph (3, 4)-coloring A generic way to characterize hardness reductions from other PCSPs (Appendix B)

• approximate (3, 5)-coloring is NP-hard via an algebraic

reduction from approximate 3-uniform hypergraph coloring

Tractability

Algebraic characterization of the power of some CSP algorithms including arc consistency and basic LP relaxation.

Appendix A: Hardness from Gap Label Cover

The mother of all inapproximability results

GapLabelCover(C, ǫ) C colors, ǫ > 0

• Input: A bipartite graph G = U ∪ V ; E and a set of

constraint functions σuv : C → C for every edge uv ∈ E

• A coloring λ : G → C satisfies an edge if σuv(λ(u)) = λ(v)
• Goal: distinguish between satisfiable instances and instances

where no more than ǫ|E| edges can be satisfied

A parallel repetition theorem [Raz ’95]

∀ǫ > 0 ∃C GapLabelCover(C, ǫ) is NP-hard

Algebraic Gap Label Cover

u u′ v σuv : σu′v :

• Example: U = {u, u′}, V = {v}, E = {uv, u′v}, C = {•, •, •}

Algebraic Gap Label Cover

σuv : σu′v : fu fu′ fv x y z x y z x y z x y z

• Example: U = {u, u′}, V = {v}, E = {uv, u′v}, C = {•, •, •}

⇒ the minor identity fu(x, x, y) ≈ fu′(x, y, y)

Hardness from Gap Label Cover

• A minor condition Σ is ǫ-robust if no ǫ-fraction of the

identities is trivial

• If ∃ǫ > 0 s.t. Pol(A, B) satisfies no ǫ-robust minor condition,

then PCSP(A, B) is NP-hard

• Corollary: If Pol(A, B) has bounded essential arity, then

PCSP(A, B) is NP-hard [Austrin, Guruswami, H˚

astad ’17]

• A combinatorial fact: Pol(K3, K4) has essential arity 1

[Brakensiek, Guruswami ’16]

• But Pol(K3, K5) has unbounded essential arity

⇒ we need a better source of hardness

Appendix B: Hardness from other PCSPs

Hardness of (3, 5)-coloring

• Source of hardness: approx. 3-uniform hypergraph coloring

PCSP(H2, HK) is NP-hard for K ≥ 2 [Dinur, Regev, Smyth ’02]

• The “Olˇ

s´ ak” minor condition (note Pol(H2, HK) | = O): t(x, y) ≈ o(x, x, y, y, y, x) t(x, y) ≈ o(x, y, x, y, x, y) t(x, y) ≈ o(y, x, x, x, y, y)

• Claim 1: Pol(K3, K5) |

= O

• Claim 2: M |

= O ⇔ M → Pol(H2, HK) for some K ≥ 2

• Corollary: PCSP(K3, K5) is NP-hard!
• Pol(K3, K6) |

= O so we need an even better hardness result. . .

Claim 1: Pol(K3, K5) fails Olˇ s´ ak

• Construct G from K36 as “K36/O”, i.e. glue triples of the form

(x, x, y, y, y, x, x), (x, y, x, y, x, y), and (y, x, x, x, y, y)

• Easy to see: Pol(K3, K5) |

= O ⇔ G → K5

• Below is a 6-clique in G

100 011 = 010 101 = 001 110 121 212 = 112 221 = 211 122 220 002 = 022 200 = 202 020 012 120 120 201 201 012

Claim 2: M | = O ⇔ M → Pol(H2, HK) for some K ≥ 2

⇐ easy since Pol(H2, HK) | = O ⇒ a bit of “abstract nonsense”: the free structure F = FM(H2)

• vertices: binary (|H2|-ary) functions from M
• hyperedges: (f , g, h) such that ∃o′ ∈ M

f (x, y) ≈ o′(x, x, y, y, y, x) g(x, y) ≈ o′(x, y, x, y, x, y) h(x, y) ≈ o′(y, x, x, x, y, y)

• M |

= O ⇒ no constant hyperedge ⇒ colorable by K = |F| colors ⇒ F → HK

• A universal property of free structures:

FM(A) → B ⇔ M → Pol(A, B)