Random 2-XORSAT/MAX-2-XORSAT and their phase transitions V LADY R AVELOMANANA LIAFA – UMR CNRS 7089 . Université Denis Diderot. vlad@liafa.jussieu.fr joint work with H ERVÉ D AUDÉ (LATP – Université de Provence) & V ONJY R ASENDRAHASINA (LIPN – Université de Paris-Nord) Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 1 / 44
Outline of the talk Introduction & motivations. The 2-XORSAT phase transition. MAX-2-XORSAT. Conclusion and perspectives. Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 2 / 44
Introduction & Motivations Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 3 / 44
Decision and optimization problems Decision and optimization problems play central key rôle in CS (cf. [G AREY , J OHNSON 79] , [A USIELLO et al. 03] ) A decision problem is a question in some formal system with a 1 yes / no answer : INPUT : an instance I and a property P . OUTPUT : yes or no I satisfies P . An optimization problem is the problem of finding the best 2 solution from all feasible solutions. In this talk, we consider two such problems : 2-XORSAT and MAX-2-XORSAT . Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 4 / 44
SAT-like problems Random k -SAT formulas ( k > 2) are subject to phase transition phenomena [F RIEDGUT , B OURGAIN 1999] . Main research tasks include Localization of the threshold (ex. 3-SAT 4.2. . . ? 1 3-XORSAT 0.91. . . [D UBOIS , M ANDLER 03)] ) Nature of the phenomena : sharp/coarse . 2 [C REIGNOU , D AUDÉ 2000++] . Details inside the window of transition 3 (ex. 2-SAT [B OLLOBÀS , B ORGS , K IM , W ILSON 01] ) Space of solutions (ex. [A CHLIOPTAS , N AOR , P ERES 07] or 4 [M ONASSON et al. 07] ) Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 5 / 44
SAT-like problems : localization of 2-SAT’s threshold An instance : ( v 1 ∨ v 2 ) ∧ ( ¬ v 1 ∨ v 3 ) ∧ ( ¬ v 1 ∨ ¬ v 2 ) A solution : SAT with ( v 1 = 1 , v 2 = 0 , v 3 = 1 ) . Localization of the threshold : n variables, m = c × n clauses � n � randomly picked from the set of 4 clauses. 2 c < 1 Proba SAT → 1, c > 1 Proba SAT → 0. Underlying combinatorial structures : directed graphs . � ¬ x = 1 = ⇒ y = 1 x ∨ y Write as ¬ y = 1 = ⇒ x = 1 Characterization : SAT iff no directed path between x and ¬ x (and vice-versa). Proof. First and second moments method [G OERDT 92, D E LA V EGA 92, C HVÀTAL , R EED 92] . Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 6 / 44
2 -XORSAT / MAX- 2 -XORSAT Main motivations Since the empirical results ( [K IRKPATRICK , S ELMAN 90] about k -SAT, rigorous results are quite limited ! What are the contributions of E NUMERATIVE /A NALYTIC C OMBINATORICS to SAT/CSP-like problems? M ONASSON (2007) inferred that (statistical physics) : � � 2XORSAT ( n , n n → + ∞ n critical exponent × Proba lim = O ( 1 ) , 2 ) where “critical exponent” = 1 / 12 . We will show that “critical exponent” = 1 / 12 and will explicit the hidden constant behind the O ( 1 ) . We will quantify the MAXIMUM number of satisfiable clauses in random formula. Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 7 / 44
The 2-XORSAT phase transition Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 8 / 44
Random 2 -XORSAT Ex : x 1 ⊕ x 2 = 1 , x 2 ⊕ x 3 = 0 , x 1 ⊕ x 3 = 0 , x 3 ⊕ x 4 = 1 , · · · . General form : AX = C where A has m rows and 2 columns and C is a m -dimensional 0 / 1 vector. Distribution : uniform. We pick m clauses of the form x i ⊕ x j = ε ∈ { 0 , 1 } from the set of n ( n − 1 ) clauses. Underlying structures : graphs with weighted edges ⇒ edges of weight ε ∈ { 0 , 1 } . x ⊕ y = ε ⇐ Characterisation : SAT iff no elementary cycle of odd weight. Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 9 / 44
SAT iff no elementary cycle of odd weight 1 ��� ��� ��� ��� 0 ��� ��� ��� ��� 1 x 1 ⊕ x 2 = 1 3 �� �� �� �� 2 �� �� �� �� 0 x 2 ⊕ x 3 = 0 �� �� �� �� 1 x 1 ⊕ x 3 = 0 x 3 ⊕ x 4 = 1 4 �� �� �� �� �� �� UNSAT ⇐ = Fix a cycle of odd weight ... SAT ⇐ = No cycles of odd weight. DFS affectation based proof. Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 10 / 44
Main ideas of our approach A basic scheme Enumeration of “SAT”-graphs (graphs without cycles of odd 1 weight) by means of generating functions. Use the obtained results with analytic combinatorics to compute : 2 Prob. SAT = Nbr of configurations without cycles of odd weight . Nbr total of configurations Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 11 / 44
Taste of our results : the whole window 1 0,8 0,6 0,4 0,2 0 0 0,2 0,4 0,6 0,8 c def = Proba [ 2 − XOR with n variables , cn clauses ] is SAT p ( n , cn ) for n = 1000 , n = 2000 and the theoretical function : e c / 2 ( 1 − 2c ) 1 / 4 . Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 12 / 44
Taste of our results: rescaling the critical window 1,6 1,2 0,8 0,4 0 -4 -2 0 2 4 Rescaling at the point “zero”, i.e c = 1 / 2 : n = 1000 , n = 2000 and lim n →∞ n 1 / 12 × � �� � p ( n , n / 2 + µ n 2 / 3 ) as a function of µ . Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 13 / 44
Enumerating graphs of 2 -XORSAT. We will enumerate the connected graphs without cycles of odd weight according to two parameters: number of vertices n and def number of edges n + ℓ . ℓ = excess . Let z n � C ℓ ( z ) = c n , n + ℓ n ! . n > 0 What are the series C ℓ ? Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 14 / 44
Enumerating graphs of 2 -XORSAT. We will enumerate the connected graphs without cycles of odd weight according to two parameters: number of vertices n and def number of edges n + ℓ . ℓ = excess . Let z n � C ℓ ( z ) = c n , n + ℓ n ! . n > 0 What are the series C ℓ ? Th. C ℓ ( z ) = 1 2 W ℓ ( 2 z ) with W ℓ = Exponential generating functions of connected graphs W RIGHT (1977). Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 14 / 44
Enumerations: trees and unicyclic components Rooted and unrooted trees (excess = − 1) ( 2 n ) n − 1 z n � T ( z ) = ze 2 T ( z ) = C − 1 ( z ) = T − T 2 . n ! , n > 0 Unicyclic components (excess = 0) Number of labellings of a smooth cycle (i.e. without vertices of 1 degree 1) using n > 2 vertices : 2 n n ! 2 n . Thus, the EGF of smooth unicyclic components 2 C 0 ( z ) = − 1 ˜ 4 log ( 1 − 2 z ) − z / 2 − z 2 / 2 . Substituting each vertex with a full rooted tree, we get 3 C 0 ( z ) = − 1 4 log ( 1 − 2 T ) − T / 2 − T 2 / 2 . What about multicyclic components? (excess > 0) Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 15 / 44
Enumerations: connected multicyclic components 0 0 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 On a connected “SAT”-graph with n vertices and n + ℓ edges, the edges of a spanning tree can be colored in 2 n − 1 ways. The colors of the other edges are “determined”. Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 16 / 44
Enumerations: general multicyclic components Let F r ( z ) be the EGF of all complex weighted labelled graphs ( connected or not ), with a positive total excess 1 r and without cycles of odd weight (“SAT-graph”). X ! W k ( 2 z ) X F r ( z ) = exp 2 r ≥ 0 k ≥ 1 and for any r ≥ 1 r k W k ( 2 z ) X F 0 ( z ) = 1 . rF r ( z ) = F r − k ( z ) , 2 k = 1 Since W k ( x ) ≍ w k [W RIGHT 80] , we also have F k ( x ) ≍ ( 1 − T ( 2 x )) 3 r with f k ( 1 − T ( x )) 3 r 2 rf r = P r r > 0. k = 1 kb k f r − k , def 1 total excess of the random graphs = nbr of edges + number of trees − number of vertices Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 17 / 44
The Random 2-XORSAT Transition Th. The probability that a random formula with n variables and m clauses is SAT satisfies the following : (i) Sub-critical phase : As 0 < n − 2 m ≪ n 2 / 3 , „ « n 2 Pr ( n , m ) = e m / 2n “ 1 − 2m ” 1 / 4 + O . n ( n − 2m ) 3 2 + µ n 2 / 3 , µ ∈ R fixed (ii) Critical phase : As m = n “ n , n ” n →∞ n 1 / 12 Pr lim 2 ( 1 + µ n − 1 / 3 ) = Ψ( µ ) , where Ψ can be expressed in terms of the Airy function. 2 + µ n 2 / 3 with µ = o ( n 1 / 12 ) (iii) Super-critical phase : As m = n “ n , n ” = Poly ( n , µ ) e − µ 3 2 ( 1 + µ n − 1 / 3 ) 6 . Pr Vlady Ravelomanana Random (MAX)–2–XORSAT phase transitions November 30, 2009. 18 / 44
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