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 = { C 1 , . . . , C m } – finitely many constraints C i = (¯ s i , R i ) , s i is a k i -tuple of variables and R i ⊆ A k i where ¯ • Q: Is there a solution, i.e. ϕ : V → A such that ϕ (¯ s i ) ∈ R i ? NP -complete, a natural restriction: fixed template CSP s • Fix A and finitely many admissible relations R i , i.e. a finite relational structure A = ( A ; R 1 , . . . , R k ) • 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 : A n → A preserving all the constraint relations, i.e. for each R A and a i ∈ R A , f ( a 1 , . . . , a n ) ∈ R A • a multivariate endomorphism f : A n → A Key insight: “more symmetric is easier” • S ⊆ A k 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 : C 1 → C 2 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 ) → P 2 (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 ( K c , K d ) Believed NP -hard for all 3 ≤ c < d , known for • (3 , 4) [Khanna, Linial, Safra ’00] 1 3 ) ) for big enough K [Huang ’13] • ( K , 2 Ω( K • (4 , 6) , ( k , 2 k − 2) [Brakensiek, Guruswami ’16] • (3 , 5) , ( k , 2 k − 1) [JB, Krokhin, Oprˇ sal ’19]
Polymorphisms for PCSP A polymorphism of PCSP ( A , B ) • f : A n → B s.t. a 1 , . . . , a n ∈ R A ⇒ f ( a 1 , . . . , a n ) ∈ R B endohomomorphism f : A n → B • a multivariate ✘✘ ❳❳ ✘ ❳ • Pol( A 1 , B 1 ) ⊆ Pol( A 2 , B 2 ) ⇒ PCSP ( A 2 , B 2 ) ≤ L PCSP ( A 1 , B 1 ) [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 ( x 1 , . . . , x m ) = f ( x σ (1) , . . . , x σ ( m ) ) , for any σ : [ m ] → [ n ]
Abstract minions • minor homomorphism = Φ : M 1 → M 2 preserving minors • minor condition = a finite set of minor identities: g ( x 1 , . . . , x m ) ≈ f ( x σ (1) , . . . , x σ ( 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( A 2 , B 2 ) satisfies all minor conditions satisfied by Pol( A 1 , B 1 ) , then PCSP ( A 2 , B 2 ) ≤ L PCSP ( A 1 , B 1 ) . • for CSPs partly already in [Barto, Oprˇ sal, Pinsker ’18] • a completely new, straightforward proof
Sketch of proof An intermediate problem: promise minor condition PMC M ( N ) • Input: a minor condition Σ in at most N variables • ACCEPT if Σ is trivial, REJECT if M �| = Σ 1 PCSP ( A 2 , B 2 ) ≤ L PMC Pol( A 2 , B 2 ) ( N ) for N = max R {| A 2 | , | R A 2 | } • a construction: from X to Σ (on the next slide) • X → A 2 ⇒ Σ trivial • Pol( A 2 , B 2 ) | = Σ ⇒ X → B 2 2 PMC Pol( A 2 , B 2 ) ( N ) ≤ L PMC Pol( A 1 , B 1 ) ( N ) for all N > 0 • trivially from the assumption 3 PMC Pol( A 1 , B 1 ) ( N ) ≤ L PCSP ( A 1 , B 1 ) for all N > 0 • “indicator construction”: variables f ( a 1 , . . . , a n ) , assert that f ’s are polymorphisms, add = where required by the identities
Example of construction • Let A 2 = �{ 0 , 1 } ; NAE � • Function symbols: • f x of arity | A 2 | = 2 for x ∈ X , • g C of arity | R A 2 | = 6 for every constraint C : ( x , y , z ) ∈ R • Identities:for every constraint C : ( x , y , z ) ∈ R add f x ( u 0 , u 1 ) ≈ g C ( u 0 , u 0 , u 1 , u 1 , u 1 , u 0 ) f y ( u 0 , u 1 ) ≈ g C ( u 0 , u 1 , u 0 , u 1 , u 0 , u 1 ) f z ( u 0 , u 1 ) ≈ g C ( u 1 , u 0 , u 0 , u 0 , u 1 , u 1 ) • X → A 2 ⇒ Σ trivial: f x := projection to ϕ ( x ) th coordinate • Pol( A 2 , B 2 ) | = Σ ⇒ X → B 2 : ϕ ( x ) := f x (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 σ uv : C → C for every edge uv ∈ E • A coloring λ : G → C satisfies an edge if σ uv ( 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 • • σ uv : • • u • • v • • u ′ σ u ′ v : • • • • • Example: U = { u , u ′ } , V = { v } , E = { uv , u ′ v } , C = {• , • , •}
Algebraic Gap Label Cover x x σ uv : y y z z f u f v f u ′ x x σ u ′ v : y y z z • Example: U = { u , u ′ } , V = { v } , E = { uv , u ′ v } , C = {• , • , •} ⇒ the minor identity f u ( x , x , y ) ≈ f u ′ ( 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( K 3 , K 4 ) has essential arity 1 [Brakensiek, Guruswami ’16] • But Pol( K 3 , K 5 ) 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 ( H 2 , H K ) is NP -hard for K ≥ 2 [Dinur, Regev, Smyth ’02] • The “Olˇ s´ ak” minor condition (note Pol( H 2 , H K ) �| = 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( K 3 , K 5 ) �| = O • Claim 2: M �| = O ⇔ M → Pol( H 2 , H K ) for some K ≥ 2 • Corollary: PCSP ( K 3 , K 5 ) is NP -hard! • Pol( K 3 , K 6 ) | = O so we need an even better hardness result. . .
Claim 1: Pol( K 3 , K 5 ) fails Olˇ s´ ak • Construct G from K 3 6 as “ K 3 6 / 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( K 3 , K 5 ) | = O ⇔ G → K 5 • Below is a 6-clique in G 100 011 = 010 101 = 001 110 012 120 121 212 = 112 221 = 211 122 201 012 120 201 220 002 = 022 200 = 202 020
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