Contradictory circuits of the 2-SAT Sergey Dovgal LIPN, Universit´ e Paris 13 ALEA Marseille, 18/03/2019
Introduction and Context
The 2-SAT problem The 2-SAT problem is the problem of satisfiability of a 2-CNF. Example: x 1 , . . . , x 4 ∈ { 0 , 1 } . x 1 ∨ x 2 = 1 , x 1 ∨ x 4 = 1 , x 2 ∨ x 3 = 1 , x 2 ∨ x 4 = 1 , x 3 ∨ x 4 = 1 Each disjunction ( x i ∨ x j ) is called a clause .
Implication digraph 1 1 To each clause ( x ∨ y ) we assign two implica- tions: x → y 2 2 ⇔ x ∨ y . y → x 3 3 We write x � y if x implies y , i.e. if there exists a path from x to y . 4 4 Lemma. Formula is unsatisfiable iff there exists a contradictory variable x : x � x � x
Contradictory component Contradictory component is the set of contradictory vertices Contradictory component ( F ) = { x | x � x � x in F } Lemma. Contradictory component is a set of strongly connected components. There are no implication paths between them.
The Spine The Spine is the set of vertices forced to take the FALSE value Spine ( F ) = { x | x � x in F }
The 2-SAT Summary
Tree-like structures A tree is a connected graph with k vertices and k − 1 edges. A unicycle is a connected graph with k vertices and k edges. A bicycle is a connected graph with k vertices and k + 1 edges. 3 2 3 3 1 6 2 1 3 2 2 1 4 4 4 1 4 5 − 1 0 1 2
Random graphs 3 2 3 3 1 6 2 1 3 2 2 1 4 4 4 1 4 5 − 1 0 1 2 The excess of a connected component is equal to # of edges minus # of vertices. The complex component of a graph is the set of components with positive excess.
Phase transition in 2-SAT and random graphs
Phase transition in random graphs G ( n , m ) : a random graph with n vertices and m edges. Critical range: m = n 2 ( 1 + µ n − 1 / 3 ) 1 − 5 1 | µ | 3 + · · · , µ → −∞ ; 24 � � G ( n , m ) contains P ( µ ) , µ = Θ( 1 ); P ∼ no complex component √ e − µ 3 / 6 2 π µ → + ∞ µ 3 / 4 , 2 1 / 4 Γ( 1 / 4 ) 3 2 3 3 1 6 2 1 3 2 2 1 4 4 4 1 4 5 − 1 0 1 2
Phase transition in random 2-CNF F ( n , m ) : formulae with n variables and m clauses. Critical range: m = n ( 1 + µ n − 1 / 3 ) 1 − Θ( 1 | µ | 3 ) , µ → −∞ ; � � P F ( n , m ) is SAT ∼ Θ( 1 ) , µ = Θ( 1 ); exp( − Θ( µ 3 )) , µ → + ∞
Motivations behind the 2-SAT phase transition 1. Give a missing analytic description and obtain more precise results for the transition curve 2. 2-SAT is an interpolation between graphs and directed graphs 3. A similar challenging problem: (giant) strongly connected component in critical directed graphs
In the backstage 1. Previous related work: (small) subgraphs in random graphs [Collet, de Panafieu, Gardy, Gitenberger, Ravelomanana] 2. This talk: sum-representation technique. ◮ Less powerful technique, but gives immediate result. 3. A more powerful approach: ongoing work with ´ Elie de Panafieu and Vlady Ravelomanana . ◮ Analytic descriptions are already available ◮ Asymptotic analysis to be done.
Short announcement of the ongoing work
Ongoing work announcement ♥ [de Panafieu, D., ’19] Let a n , m be the # of graphs from a family F 2 with n vertices and m edges. The exponential w 5 z 4 generating function of F is defined as 3 4 ! z n w m � F ( z , w ) := a n , m . 4 1 n ! n , m � 0 Theorem ♥ . The EGF S ( z , w ) for strongly connected directed graphs and the EGF G ( z , w ) for simple graphs satisfy the relation � 1 � S ( z , w ) = − log G ( z , w ) ⊙ z G ( z , w ) where ⊙ z is the exponential Hadamard product a n ( w ) z n b n ( w ) z n a n ( w ) b n ( w ) z n � � � . ⊙ z = n ! n ! n ! n n n
The results
The results Subcritical phase: m = n ( 1 + µ n − 1 / 3 ) , µ → −∞ . Theorem. As µ → −∞ , P ( SAT ) has a full asymptotic expansion in powers of | µ | − 3 with explicit computable coefficients. P ( F ( n , m ) is SAT ) ∼ 1 − a 1 | µ | 3 + a 2 | µ | 6 − · · · Theorem. The number of contradictory variables follows the Gamma ( 2 ) law with density f ( x ) ∼ xe − µ n − 1 / 3 x . Theorem. Spine structure: most of the spine variables belong to disjoint tree-like spine structures
The techniques
Sum-representation Lemma. For a given formula F , # of sum-representations = 2 # of clauses − # of multiple edges The multiple edges correspond to the clauses of type ( x ∨ x ) .
Graph rotations 2 2 1 1 1 2 2 1 + = 2 2 3 3 3 3 3 3 1 1 Rotate ( x 1 ∨ x 3 ) 2 2 1 1 1 2 2 1 2 + = 2 3 3 3 3 3 3 1 1
Cherry-picking the sum-representation 5 4 4 5 3 2 2 3 1 1 1 1 1 1 = + 3 5 4 2 3 2 4 3 2 4 5 5 5 2 3 4 P ( sum-representation contains the contradictory circuit of length ℓ ) = 2 − ℓ
Typical contradictory components Subcritical phase, m = n ( 1 + µ n − 1 / 3 ) , µ → −∞ Lemma. The contribution of a contra- dictory cubic component with excess 2 r is Θ( | µ | − 3 r ) . Proof. ◮ Step 1. Cherry-pick a sum-representation (spanning tree) ◮ Step 2. Assemble the generating functions ◮ Step 3. Saddle-point asymptotics
Typical spine components Subcritical phase, m = n ( 1 + µ n − 1 / 3 ) , µ → −∞ Theorem. 2 n 2 / 3 µ − 2 vertices belonging to the tree-like spine ◮ There are 1 components. ◮ Further contributions are c 1 n 2 / 3 µ − 5 , c 2 n 2 / 3 µ − 8 , .... ◮ Further contributions are obtained by increasing the complexity of the spine component
Spine components of increased complexity
Limitations of the inclusion-exclusion Inside the critical window, m = n ( 1 + µ n − 1 / 3 ) , µ = constant Let ξ be a discrete random variable ξ ∈ { 0 , 1 , 2 , 3 , · · · } . Then, P ( ξ = 0 ) = 1 − E ξ + E ξ ( ξ − 1 ) − E ξ ( ξ − 1 )( ξ − 2 ) + · · · 2 ! 3 ! The same event A can be expressed as [ ξ = 0 ] for different possible ξ . ◮ P ( F ( n , m ) is SAT ) = P ( ξ 1 = 0 ) = P ( ξ 2 = 0 ) ◮ ξ 1 = number of contradictory variables ◮ ξ 2 = number of contradictory components Challenge: multiple counting of the overlapping contradictory components gives divergent series for E ξ 2 → + ∞ when µ � µ c , µ c < 0 .
Conclusion
Conclusion 1. 2-SAT forms a synergy between random graphs and random directed graphs 2. Sum-representation approach identifies the analog of the complex component in a random 2-SAT, and the role of cubic graphs 3. The sum-representation subgraph approach is limited: a more precise inclusion-exclusion is required for the full range
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