Constraints, Symmetry, and Complexity Part 2 Andrei Krokhin Andrei - PowerPoint PPT Presentation

Constraints, Symmetry, and Complexity Part 2 Andrei Krokhin Andrei Krokhin Constraints, Symmetry, and Complexity

Mantra Symmetries of solution spaces are relevant for complexity Lack of symmetries ⇒ hardness Symmetries ⇒ efficient algorithms Symmetries of higher dimension/arity are important Andrei Krokhin Constraints, Symmetry, and Complexity

Motivational example: Approximate graph colouring Classical fact: For any fixed k ≥ 3, it’s NP -hard to find a k -colouring of a given k -colourable graph. Andrei Krokhin Constraints, Symmetry, and Complexity

Motivational example: Approximate graph colouring Classical fact: For any fixed k ≥ 3, it’s NP -hard to find a k -colouring of a given k -colourable graph. Natural question: How many colours are needed to efficiently find a colouring? Andrei Krokhin Constraints, Symmetry, and Complexity

Motivational example: Approximate graph colouring Classical fact: For any fixed k ≥ 3, it’s NP -hard to find a k -colouring of a given k -colourable graph. Natural question: How many colours are needed to efficiently find a colouring? Old (in)famous problem: for any fixed const 3 ≤ k ≤ c , is it NP -hard to find a c -colouring of a given k -colourable graph? Andrei Krokhin Constraints, Symmetry, and Complexity

Motivational example: Approximate graph colouring Classical fact: For any fixed k ≥ 3, it’s NP -hard to find a k -colouring of a given k -colourable graph. Natural question: How many colours are needed to efficiently find a colouring? Old (in)famous problem: for any fixed const 3 ≤ k ≤ c , is it NP -hard to find a c -colouring of a given k -colourable graph? 3 vs. 4 colouring is NP -hard [GJ76], but 3 vs. 5 is open. Andrei Krokhin Constraints, Symmetry, and Complexity

Motivational example: Approximate graph colouring Classical fact: For any fixed k ≥ 3, it’s NP -hard to find a k -colouring of a given k -colourable graph. Natural question: How many colours are needed to efficiently find a colouring? Old (in)famous problem: for any fixed const 3 ≤ k ≤ c , is it NP -hard to find a c -colouring of a given k -colourable graph? 3 vs. 4 colouring is NP -hard [GJ76], but 3 vs. 5 is open. k vs. c = 2 k − 2 is NP -hard [Brakensiek, Guruswami’16] Andrei Krokhin Constraints, Symmetry, and Complexity

Motivational example: Approximate graph colouring Classical fact: For any fixed k ≥ 3, it’s NP -hard to find a k -colouring of a given k -colourable graph. Natural question: How many colours are needed to efficiently find a colouring? Old (in)famous problem: for any fixed const 3 ≤ k ≤ c , is it NP -hard to find a c -colouring of a given k -colourable graph? 3 vs. 4 colouring is NP -hard [GJ76], but 3 vs. 5 is open. k vs. c = 2 k − 2 is NP -hard [Brakensiek, Guruswami’16] k vs. 2 Ω( k 1 / 3 ) is NP -hard for large enough k [Huang’13] Andrei Krokhin Constraints, Symmetry, and Complexity

Motivational example: Approximate graph colouring Classical fact: For any fixed k ≥ 3, it’s NP -hard to find a k -colouring of a given k -colourable graph. Natural question: How many colours are needed to efficiently find a colouring? Old (in)famous problem: for any fixed const 3 ≤ k ≤ c , is it NP -hard to find a c -colouring of a given k -colourable graph? 3 vs. 4 colouring is NP -hard [GJ76], but 3 vs. 5 is open. k vs. c = 2 k − 2 is NP -hard [Brakensiek, Guruswami’16] k vs. 2 Ω( k 1 / 3 ) is NP -hard for large enough k [Huang’13] NP -hard for all 3 ≤ k ≤ c , assuming non-standard (perfect completeness) variants of the UGC [Dinur, Mossel, Regev’09] Andrei Krokhin Constraints, Symmetry, and Complexity

Approximating CSP Goal: Given a CSP instance, find an approximate solution. Options: Andrei Krokhin Constraints, Symmetry, and Complexity

Approximating CSP Goal: Given a CSP instance, find an approximate solution. Options: 1. Can violate some constraints, try to satisfy as many as poss — Full understanding modulo UGC [Raghavendra’08] Andrei Krokhin Constraints, Symmetry, and Complexity

Approximating CSP Goal: Given a CSP instance, find an approximate solution. Options: 1. Can violate some constraints, try to satisfy as many as poss — Full understanding modulo UGC [Raghavendra’08] 2. Relax the constraints themselves ( this talk ) Andrei Krokhin Constraints, Symmetry, and Complexity

Approximating CSP Goal: Given a CSP instance, find an approximate solution. Options: 1. Can violate some constraints, try to satisfy as many as poss — Full understanding modulo UGC [Raghavendra’08] 2. Relax the constraints themselves ( this talk ) 3. Perhaps do both simultaneously — some time in the future Andrei Krokhin Constraints, Symmetry, and Complexity

Promise CSP (PCSP) Fix structures A = ( A ; R 1 , . . . , R n ) and B = ( B ; S 1 , . . . , S n ) s.t. 1. arity ( R i ) = arity ( S i ) for all i Andrei Krokhin Constraints, Symmetry, and Complexity

Promise CSP (PCSP) Fix structures A = ( A ; R 1 , . . . , R n ) and B = ( B ; S 1 , . . . , S n ) s.t. 1. arity ( R i ) = arity ( S i ) for all i 2. ∃ h : A → B (homomorphism, i.e. h ( R i ) ⊆ S i ) — special case: A ⊆ B , h ( x ) = x , and then R i ⊆ S i . Andrei Krokhin Constraints, Symmetry, and Complexity

Promise CSP (PCSP) Fix structures A = ( A ; R 1 , . . . , R n ) and B = ( B ; S 1 , . . . , S n ) s.t. 1. arity ( R i ) = arity ( S i ) for all i 2. ∃ h : A → B (homomorphism, i.e. h ( R i ) ⊆ S i ) — special case: A ⊆ B , h ( x ) = x , and then R i ⊆ S i . Consider pairs of CSP instances: Φ strict = R 1 ( x , y , z ) , R 1 ( z , y , w ) , R 2 ( z ) , R 3 ( x , w ) , R 3 ( y , y ) Φ relaxed = S 1 ( x , y , z ) , S 1 ( z , y , w ) , S 2 ( z ) , S 3 ( x , w ) , S 3 ( y , y ) Andrei Krokhin Constraints, Symmetry, and Complexity

Promise CSP (PCSP) Fix structures A = ( A ; R 1 , . . . , R n ) and B = ( B ; S 1 , . . . , S n ) s.t. 1. arity ( R i ) = arity ( S i ) for all i 2. ∃ h : A → B (homomorphism, i.e. h ( R i ) ⊆ S i ) — special case: A ⊆ B , h ( x ) = x , and then R i ⊆ S i . Consider pairs of CSP instances: Φ strict = R 1 ( x , y , z ) , R 1 ( z , y , w ) , R 2 ( z ) , R 3 ( x , w ) , R 3 ( y , y ) Φ relaxed = S 1 ( x , y , z ) , S 1 ( z , y , w ) , S 2 ( z ) , S 3 ( x , w ) , S 3 ( y , y ) Because of h , have Φ strict is sat ⇒ Φ relaxed is sat Andrei Krokhin Constraints, Symmetry, and Complexity

Promise CSP (PCSP) Definition ( PCSP ( A , B ), Decision version) Accept if Φ strict is sat, reject if Φ relaxed is unsat. Andrei Krokhin Constraints, Symmetry, and Complexity

Promise CSP (PCSP) Definition ( PCSP ( A , B ), Decision version) Accept if Φ strict is sat, reject if Φ relaxed is unsat. Obvious facts: For any A , PCSP ( A , A ) = CSP ( A ) CSP ( A ) or CSP ( B ) is in P ⇒ PCSP ( A , B ) is in P . Andrei Krokhin Constraints, Symmetry, and Complexity

Promise CSP (PCSP) Definition ( PCSP ( A , B ), Decision version) Accept if Φ strict is sat, reject if Φ relaxed is unsat. Obvious facts: For any A , PCSP ( A , A ) = CSP ( A ) CSP ( A ) or CSP ( B ) is in P ⇒ PCSP ( A , B ) is in P . Definition ( PCSP ( A , B ), Search version) Given a satisfiable Φ strict , find a solution for Φ relaxed . Andrei Krokhin Constraints, Symmetry, and Complexity

Promise CSP (PCSP) Definition ( PCSP ( A , B ), Decision version) Accept if Φ strict is sat, reject if Φ relaxed is unsat. Obvious facts: For any A , PCSP ( A , A ) = CSP ( A ) CSP ( A ) or CSP ( B ) is in P ⇒ PCSP ( A , B ) is in P . Definition ( PCSP ( A , B ), Search version) Given a satisfiable Φ strict , find a solution for Φ relaxed . Remark: there is an obvious reduction from decision to search. Andrei Krokhin Constraints, Symmetry, and Complexity

Promise CSP (PCSP) Definition ( PCSP ( A , B ), Decision version) Accept if Φ strict is sat, reject if Φ relaxed is unsat. Obvious facts: For any A , PCSP ( A , A ) = CSP ( A ) CSP ( A ) or CSP ( B ) is in P ⇒ PCSP ( A , B ) is in P . Definition ( PCSP ( A , B ), Search version) Given a satisfiable Φ strict , find a solution for Φ relaxed . Remark: there is an obvious reduction from decision to search. Conjecture The search and the decision problems are always equivalent. Andrei Krokhin Constraints, Symmetry, and Complexity

Example: Approximate hypergraph colouring Recall k -NAE : A = ([ k ]; { ( a , b , c ) ∈ [ k ] 3 | a � = b ∨ a � = c ∨ b � = c } ) Essentially, this is k -colouring for 3-uniform hypergraphs. Andrei Krokhin Constraints, Symmetry, and Complexity

Example: Approximate hypergraph colouring Recall k -NAE : A = ([ k ]; { ( a , b , c ) ∈ [ k ] 3 | a � = b ∨ a � = c ∨ b � = c } ) Essentially, this is k -colouring for 3-uniform hypergraphs. Natural to consider ( k -NAE, c -NAE). Andrei Krokhin Constraints, Symmetry, and Complexity

Example: Approximate hypergraph colouring Recall k -NAE : A = ([ k ]; { ( a , b , c ) ∈ [ k ] 3 | a � = b ∨ a � = c ∨ b � = c } ) Essentially, this is k -colouring for 3-uniform hypergraphs. Natural to consider ( k -NAE, c -NAE). Theorem (Dinur, Regev, Smyth’05) For any 2 ≤ k ≤ c, PCSP(k-NAE, c-NAE) is NP -hard. Andrei Krokhin Constraints, Symmetry, and Complexity

Example: (1-in-3, NAE)- Sat 1-in-3 is R 1 / 3 = { (0 , 0 , 1) , (0 , 1 , 0) , (1 , 0 , 0) } NAE is 2-NAE as in hypergraph 2-colouring Andrei Krokhin Constraints, Symmetry, and Complexity