From Ramsey to Ehrenfeucht: a reduction between games Oleg - PowerPoint PPT Presentation

From Ramsey to Ehrenfeucht: a reduction between games Oleg Verbitsky Humboldt Universit¨ at IAPMM and Berlin, Germany Lviv, Ukraine Bertinoro, October 2009 Joint work with Frank Harary and Wolfgang Slany.

Basics of Graph Ramsey theory Definition. G → F if, for any coloring of E ( G ) in red and blue, G contains a monochromatic copy of F . Ramsey theorem. There is a function N = N ( n ) such that K N → K n (and hence K N → F for any F on n vertices). Burr (Garey and Johnson GT6): Deciding if G → K 3 is coNP-complete. 1

Ramsey games on ( G, F ) A and B color E ( G ) alternately, one edge per move A in red, B in blue A moves first Player’s objective in ACHIEVE ( G, F ) : create a monochromatic F AVOID ( G, F ) : avoid such an F Strong version: A and B have the same objective. Observation: If G → F , then the game never ends in a draw! Weak version: A has the objective, B plays against (most studied but out the scope of this talk). 2

Example. AVOID ( K 6 , K 3 )= SIM Mead, Rosa, Huang 74: SIM is won by B Open question (J´ ozsef Beck 08). Who wins AVOID ( K 18 , K 4 ) ? 3

Symmetry breaking-preserving game Rules of SYM ( G ) : A round: A ’ move + B ’s move Objective of B : to keep the red and the blue subgraphs of G isomorphic after each round Observation: If B wins SYM ( G ) , then he does not lose AVOID ( G, F ) for any F . 4

Mirror strategy in SYM ( G ) B wins SYM ( G ) whenever G has a good automorphism. An automorphism is good if it is involutory and leaves no edge fixed. C auto denotes the class of graphs with a good automorphism. C auto includes • Paths and cycles of even length. • Platonic graphs except the tetrahedron. • Cubes. • K s,t if st is even. 5

Mirror strategy in SYM ( G ) B wins SYM ( G ) whenever G has a good automorphism. An automorphism is good if it is involutory and leaves no edge fixed. C auto denotes the class of graphs with a good automorphism. C auto is closed with respect to the • sum • Cartesian, lexicographic, categorical products C auto is NP-complete. 6

Length of the game L sym ( G ) = max k s.t. B wins the k -round SYM ( G ) . Known: • L sym ( K n ) ≤ 6 • L sym ( G ) = | E ( G ) | / 2 if G ∈ C auto . In particular, – L sym ( P n ) = L sym ( C n ) = n/ 2 if n is even, where P n (resp. C n ) denotes the path (resp. cycle) of length n . – L sym ( K n,n ) = n 2 / 2 if n is even • n − 1 ≤ L sym ( K n,n ) ≤ 2 n + 38 if n is odd (Pikhurko 03) 2 7

Length of the game L sym ( G ) = max k s.t. B wins the k -round SYM ( G ) . Theorem. If n is odd, then 1. L sym ( P n ) = Ω(log n ) and L sym ( C n ) = Ω(log n ) , 2. L sym ( P n ) = O (log 2 n ) and L sym ( C n ) = O (log 2 n ) . 8

Lower bound: a connection to the Ehrenfeucht game Rules of EF ( G 0 , G 1 ) , the Ehrenfeucht-Fra ¨ ıss´ e game on graphs G 0 and G 1 Players: Spoiler Duplicator i -th round: Spoiler selects u i ∈ V ( G a ) Duplicator selects v i ∈ V ( G 1 − a ) Duplicator’s objective: to keep the correspondence ‘ u i ↔ v i ’ being a partial isomorphism between G 0 and G 1 . L EF ( G 0 , G 1 ) = max k s.t. B wins the k -round EF ( G 0 , G 1 ) . 9

Lower bound: a connection to the Ehrenfeucht game Ehrenfeucht’s theorem. No first order sentence of quantifier depth L EF ( G 0 , G 1 ) distinguishes between non-isomorphic G 0 and G 1 . On the other hand, depth L EF ( G 0 , G 1 ) + 1 suffices. Theorem (textbooks in Finite Model Theory). For every n , 1. log n − 2 < L EF ( P n , P n +1 ) < log n + 2 . 2. log n − 1 < L EF ( C n , C n +1 ) < log n + 1 . 10

Proof of the lower bound L sym ( C n ) ≥ 1 4 log n − 1 4 for odd n . “ L sym ( G ) ≥ k ” is expressible by a first order sentence Φ k with 4 k quantifiers. Let k = ⌈ log n − 1 ⌉ . 4 Since C n +1 ∈ C auto , we have C n +1 | = Φ k . Since L EF ( C n , C n +1 ) > log n − 1 , we have C n | = Φ k too. 11

Constructivization? EF ( C n , C n +1 ) ր ց SYM ( C n ) SYM ( C n +1 ) Question: We know a strategy for B in SYM ( C n +1 ) . Can we know it in SYM ( C n ) ? Answer: Yes, because we know Duplicator’s strategy in EF ( C n , C n +1 ) ! 12

Preliminaries: the line graph L ( H ) denotes the line graph of a graph H : V ( L ( H )) = E ( H ) , e 1 and e 2 are adjacent in L ( H ) if they have a common vertex in H . Example: L ( C n ) = C n , L ( P n ) = P n − 1 Clearly, H 1 ∼ = H 2 ⇒ L ( H 1 ) ∼ = L ( H 2 ) . The Whitney theorem. L ( H 1 ) ∼ = L ( H 2 ) ⇒ H 1 ∼ = H 2 for all connected H 1 and H 2 unless { H 1 , H 2 } = { K 3 , K 1 , 3 } . 13

Constructivization! Our former approach generalizes to  ff L sym ( G 0 ) , 1 L sym ( G 1 ) ≥ min 4 L EF ( G 0 , G 1 ) Now we prove: If G 1 is triangle-free, then  L sym ( G 0 ) , 1 ff L sym ( G 1 ) ≥ min 2 L EF ( L ( G 0 ) , L ( G 1 )) In particular, L sym ( C n ) ≥ 1 2 log n − 1 2 . 14

Reduction Let S 0 denote a strategy of B in SYM ( G 0 ) . Let D denote a strategy of Duplicator in EF ( L ( G 0 ) , L ( G 1 )) . We describe S 1 = S 1 ( S 0 , D ) , a strategy for B in SYM ( G 1 ) , such that if S 0 succeeds in k rounds of SYM ( G 0 ) and D in 2 k rounds of EF ( L ( G 0 ) , L ( G 1 )) , then S 1 succeeds in k rounds of SYM ( G 1 ) . 15

A round of SYM ( G 1 ) EF board (G ) (G ) L L 1 0 SYM boards 1. A ’s move in SYM ( G 1 ) 2. Spoiler’s move in EF ( G 1 , G 0 ) G G 0 (simulation) 1 16

A round of SYM ( G 1 ) EF board (G ) (G ) L L 1 0 1. A ’s move in SYM ( G 1 ) 2. Spoiler’s move in EF ( G 1 , G 0 ) SYM boards (simulation) 3. Duplicator’s move in EF ( G 1 , G 0 ) (according to D ) 4. A ’s move in SYM ( G 0 ) G G 0 (simulation) 1 17

A round of SYM ( G 1 ) EF board (G ) (G ) L L 1 0 1. A ’s move in SYM ( G 1 ) 2. Spoiler’s move in EF ( G 1 , G 0 ) (simulation) 3. Duplicator’s move in EF ( G 1 , G 0 ) (according to D ) 4. A ’s move in SYM ( G 0 ) SYM boards (simulation) 5. B ’s move in SYM ( G 0 ) (according to S 0 ) 6. Spoiler’s move in EF ( G 1 , G 0 ) G G 0 (simulation) 1 18

A round of SYM ( G 1 ) EF board 1. A ’s move in SYM ( G 1 ) 2. Spoiler’s move in EF ( G 1 , G 0 ) (simulation) 3. Duplicator’s move in EF ( G 1 , G 0 ) (G ) (G ) L L 1 0 (according to D ) 4. A ’s move in SYM ( G 0 ) (simulation) 5. B ’s move in SYM ( G 0 ) (according to S 0 ) 6. Spoiler’s move in EF ( G 1 , G 0 ) SYM boards (simulation) 7. Duplicator’s move in EF ( G 1 , G 0 ) (according to D ) 8. B ’s move in SYM ( G 1 ) G G 0 (this defines S 0 ) 1 19

Analysis of the strategy Fix a strategy of A in SYM ( G 1 ) . Denote A i – red edges of G i colored up to the k -th round, B i – blue edges of G i colored up to the k -th round. Note that A 0 is constructed from A 1 and B 1 from B 0 . ∼ because S 0 succeeds A 0 = B 0 ⇓ ∼ L ( A 0 ) = L ( B 0 ) � � ∼ L ( G 0 )[ A 0 ] L ( G 0 )[ B 0 ] = �≀ �≀ because D succeeds ∼ L ( G 1 )[ A 1 ] = L ( G 1 )[ B 1 ] � � ∼ L ( A 1 ) L ( B 1 ) = ⇓ by Whitney’s theorem ∼ A 1 B 1 = 20

Thank you! 21