Clique is hard on average for regular resolution Ilario Bonacina, - PowerPoint PPT Presentation

Clique is hard on average for regular resolution Ilario Bonacina, UPC Barcelona Tech July 27, 2018 Oxford Complexity Day

Talk based on a joint work with: A. Atserias S. de Rezende M. Lauria J. Nordstr¨ om A. Razborov 1

Motivations • k -clique is a fundamental NP-complete problem • regular resolution captures state-of-the-art algorithms for k -clique • for k small (say k ≪ √ n ) the standard tools from proof complexity fail 2

k-clique Input: a graph G = ( V , E ) with n vertices and k ∈ N Output: yes if G contains a k -clique as a subgraph; no otherwise 3

k-clique Input: a graph G = ( V , E ) with n vertices and k ∈ N Output: yes if G contains a k -clique as a subgraph; no otherwise 3

k-clique Input: a graph G = ( V , E ) with n vertices and k ∈ N Output: yes if G contains a k -clique as a subgraph; no otherwise 3

k-clique Input: a graph G = ( V , E ) with n vertices and k ∈ N Output: yes if G contains a k -clique as a subgraph; no otherwise 3

k-clique Input: a graph G = ( V , E ) with n vertices and k ∈ N Output: yes if G contains a k -clique as a subgraph; no otherwise 3

k-clique Input: a graph G = ( V , E ) with n vertices and k ∈ N Output: yes if G contains a k -clique as a subgraph; no otherwise - k -clique can be solved in time n O ( k ) , e.g. by brute-force - k -clique is NP-complete - assuming ETH, there is no f ( k ) n o ( k ) -time algorithm for k -clique for any computable function f 3

Resolution clause 1 ∨ var clause 2 ∨ ¬ var ¬ y ∨ ¬ z ¬ x clause 1 ∨ clause 2 y ∨ ¬ w x ∨ w ¬ y ¬ x ∨ z 4

Resolution clause 1 ∨ var clause 2 ∨ ¬ var ¬ y ∨ ¬ z ¬ x clause 1 ∨ clause 2 y ∨ ¬ w x ∨ y x ∨ w ¬ y ¬ x ∨ z 4

Resolution clause 1 ∨ var clause 2 ∨ ¬ var ¬ y ∨ ¬ z ¬ x clause 1 ∨ clause 2 y ∨ ¬ w x ∨ y x ∨ w y ∨ z ¬ y ¬ x ∨ z 4

Resolution clause 1 ∨ var clause 2 ∨ ¬ var ¬ y ∨ ¬ z ¬ x clause 1 ∨ clause 2 y ∨ ¬ w x ∨ y x ∨ w y ∨ z z ¬ y ¬ x ∨ z 4

Resolution clause 1 ∨ var clause 2 ∨ ¬ var ¬ y ∨ ¬ z ¬ x clause 1 ∨ clause 2 x ∨ ¬ z y ∨ ¬ w x ∨ y x ∨ w y ∨ z z ¬ y ¬ x ∨ z 4

Resolution clause 1 ∨ var clause 2 ∨ ¬ var ¬ y ∨ ¬ z ¬ x clause 1 ∨ clause 2 x ∨ ¬ z ¬ z y ∨ ¬ w x ∨ y x ∨ w y ∨ z z ¬ y ¬ x ∨ z 4

Resolution clause 1 ∨ var clause 2 ∨ ¬ var ¬ y ∨ ¬ z ¬ x clause 1 ∨ clause 2 x ∨ ¬ z ¬ z y ∨ ¬ w x ∨ y ⊥ x ∨ w y ∨ z z ¬ y ¬ x ∨ z 4

Resolution clause 1 ∨ var clause 2 ∨ ¬ var ¬ y ∨ ¬ z ¬ x clause 1 ∨ clause 2 x ∨ ¬ z ¬ z y ∨ ¬ w x ∨ y ⊥ x ∨ w y ∨ z z ¬ y ¬ x ∨ z Tree-like = the proof DAG is a tree Regular = no variable resolved twice in any source-to-sink path Size = # of nodes in the proof DAG 4

What is Resolution good for? • algorithms routinely used to solve SAT (CDCL-solvers) are somewhat formalizable in resolution • the state-of-the-art algorithms to solve k-clique (Bron-Kerbosch, ¨ Osterg˚ ard, Russian dolls algorithms, ...) are formalizable in regular resolution 5

k-clique formula Construct a propositional formula Φ G , k unsatisfiable if and only if “ G does not contain a k -clique” x v , j ≡ “ v is the j -th vertex of a k -clique in G ”. The clique formula Φ G , k � x v , i for i ∈ [ k ] v ∈ V and ¬ x u , i ∨ ¬ x v , i for i ∈ [ k ] , u , v ∈ V and ¬ x u , i ∨ ¬ x v , j for i � = j ∈ [ k ] , u , v ∈ V , ( u , v ) / ∈ E 6

Size S (Φ G , k ) = minimum size of a resolution refutation of Φ G , k S tree (Φ G , k ) = minimum size of a tree-like resolution ref. of Φ G , k S reg (Φ G , k ) = minimum size of a regular resolution ref. of Φ G , k • S (Φ G , k ) � S reg (Φ G , k ) � S tree (Φ G , k ) � n O ( k ) • if G is ( k − 1)-colorable then S reg (Φ G , k ) � 2 k k 2 n 2 [ ∼ BGL13] [BGL13] Beyersdorff, Galesi and Lauria 2013. Parameterized complexity of DPLL search procedures. 7

Erd˝ os-R´ enyi random graphs A graph G = ( V , E ) ∼ G ( n , p ) is such that | V | = n and each edge { u , v } ∈ E independently with prob. p ∈ [0 , 1] 8

Erd˝ os-R´ enyi random graphs A graph G = ( V , E ) ∼ G ( n , p ) is such that | V | = n and each edge { u , v } ∈ E independently with prob. p ∈ [0 , 1] • if p ≪ n − 2 / ( k − 1) then a.a.s. G ∼ G ( n , p ) has no k -cliques • A.a.s. G ∼ G ( n , 1 2 ) has no clique of size ⌈ 2 log 2 n ⌉ 8

Main Result (simplified) Main Theorem (version 1) Let G ∼ G ( n , p ) be an Erd˝ os-R´ enyi random graph with, for simplicity, p = n − 4 / ( k − 1) and let k � n 1 / 2 − ǫ for some arbitrary small ǫ . Then, S reg (Φ G , k ) a.a.s. = n Ω( k ) . 9

Main Result (simplified) Main Theorem (version 1) Let G ∼ G ( n , p ) be an Erd˝ os-R´ enyi random graph with, for simplicity, p = n − 4 / ( k − 1) and let k � n 1 / 2 − ǫ for some arbitrary small ǫ . Then, S reg (Φ G , k ) a.a.s. = n Ω( k ) . the actual lower bound decreases smoothly w.r.t. p 9

Main Result (simplified) Main Theorem (version 1) Let G ∼ G ( n , p ) be an Erd˝ os-R´ enyi random graph with, for simplicity, p = n − 4 / ( k − 1) and let k � n 1 / 2 − ǫ for some arbitrary small ǫ . Then, S reg (Φ G , k ) a.a.s. = n Ω( k ) . the actual lower bound decreases smoothly w.r.t. p Main Theorem (version 2) Let G ∼ G ( n , 1 2 ), then = n Ω(log n ) for k = O (log n ) S reg (Φ G , k ) a.a.s. and = n ω (1) for k = o (log 2 n ) . S reg (Φ G , k ) a.a.s. 9

How hard is to prove that a graph is Ramsey? Open Problem Let G be a graph in n vertices with no set of k vertices forming a clique or independent set, where k = ⌈ 2 log n ⌉ . Is it true that S ( reg ) (Φ G , k ) = n Ω(log n ) ? ([LPRT17] proved this but for a binary encoding of Φ G , k ) [LPRT17] Lauria, Pudl´ ak, R¨ odl, and Thapen, 2017. The complexity of proving that a graph is Ramsey. 10

Previous lower bounds [BGL13] If G is the complete ( k − 1)-partite graph, then S tree (Φ G , k ) = n Ω( k ) . The same holds for G ∼ G ( n , p ) with suitable edge density p . [BIS07] for n 5 / 6 ≪ k < n 3 and G ∼ G ( n , p ) (with suitable edge density p ), then S (Φ G , k ) a.a.s. = 2 n Ω(1) [LPRT17] if we encode k -clique using some other propositional encodings (e.g. in binary) we get n Ω( k ) size lower bounds for resolution [BIS07] Beame, Impagliazzo and Sabharwal, 2007. The resolution complexity of independent sets and vertex covers in random graphs . [LPRT17] Lauria, Pudl´ ak, R¨ odl, and Thapen, 2017. The complexity of proving that a graph is Ramsey. 11

Rest of the talk Focus on k = ⌈ 2 log n ⌉ and G ∼ G ( n , 1 2 ), and how to prove S reg (Φ G , k ) a.a.s. = n Ω(log n ) 12

Proof scheme Theorem 1 Let k = ⌈ 2 log n ⌉ . A.a.s. G = ( V , E ) ∼ G ( n , 1 2 ) is such that: 1. V is ( k 50 , Θ( n 0 . 9 ))-dense; and k 10000 , Θ( n 0 . 9 ))-dense W ⊆ V there exists S ⊆ V , 2. For every ( | S | � √ n s.t. for every R ⊆ V , with | R | � k 50 and | � N W ( R ) | < � Θ( n 0 . 6 ) it holds that | R ∩ S | � k 10000 . 13

Proof scheme Theorem 1 Let k = ⌈ 2 log n ⌉ . A.a.s. G = ( V , E ) ∼ G ( n , 1 2 ) is such that: 1. V is ( k 50 , Θ( n 0 . 9 ))-dense; and k 10000 , Θ( n 0 . 9 ))-dense W ⊆ V there exists S ⊆ V , 2. For every ( | S | � √ n s.t. for every R ⊆ V , with | R | � k 50 and | � N W ( R ) | < � Θ( n 0 . 6 ) it holds that | R ∩ S | � k 10000 . Theorem 2 Let k = ⌈ 2 log n ⌉ . For every G satisfying properties (1) and (2), S reg (Φ G , k ) = n Ω(log n ) 13

Proof scheme Theorem 1 Let k = ⌈ 2 log n ⌉ . A.a.s. G = ( V , E ) ∼ G ( n , 1 2 ) is such that: 1. V is ( k 50 , Θ( n 0 . 9 ))-dense; and k 10000 , Θ( n 0 . 9 ))-dense W ⊆ V there exists S ⊆ V , 2. For every ( | S | � √ n s.t. for every R ⊆ V , with | R | � k 50 and | � N W ( R ) | < � Θ( n 0 . 6 ) it holds that | R ∩ S | � k 10000 . Theorem 2 Let k = ⌈ 2 log n ⌉ . For every G satisfying properties (1) and (2), S reg (Φ G , k ) = n Ω(log n ) Proof ideas: boosted Haken bottleneck counting. Bottlenecks are pair of nodes with special properties and a way of visiting them. The proof heavily uses regularity. 13

Denseness I W ⊆ V is ( r , q )-dense if for every subset R ⊆ V of size � r , it holds | � N W ( R ) | � q , where � N W ( R ) is the set of common neighbors of R in W • W • • • • • . . . • . . . • • • 14

Denseness I W ⊆ V is ( r , q )-dense if for every subset R ⊆ V of size � r , it holds | � N W ( R ) | � q , where � N W ( R ) is the set of common neighbors of R in W • W • • • • • . . . • . . . • R • • 14

Denseness I W ⊆ V is ( r , q )-dense if for every subset R ⊆ V of size � r , it holds | � N W ( R ) | � q , where � N W ( R ) is the set of common neighbors of R in W • W • • � • N W ( R ) • • . . . • . . . • R • • 14