Contradictory circuits of the 2-SAT Sergey Dovgal LIPN, Universit - - PowerPoint PPT Presentation

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: x1, . . . , x4 ∈ {0, 1}. x1 ∨ x2 = 1, x1 ∨ x4 = 1, x2 ∨ x3 = 1, x2 ∨ x4 = 1, x3 ∨ x4 = 1 Each disjunction (xi ∨ xj) is called a clause.

Implication digraph

1 2 3 4 1 2 3 4 To each clause (x ∨ y) we assign two implica- tions: x → y y → x ⇔ x ∨ y. We write x y if x implies y, i.e. if there exists a path from x to y.

• 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.

2 1 3 4 −1 4 1 3 2 1 3 4 6 5 2 1 1 2 4 3 2

Random graphs

2 1 3 4 −1 4 1 3 2 1 3 4 6 5 2 1 1 2 4 3 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 + µn−1/3) P

• G(n,m) contains

no complex component

• ∼

         1 − 5

24 1 |µ|3 + · · · ,

µ → −∞; P(µ), µ = Θ(1);

√ 2π 21/4Γ(1/4) e−µ3/6 µ3/4 ,

µ → +∞

2 1 3 4 −1 4 1 3 2 1 3 4 6 5 2 1 1 2 4 3 2

Phase transition in random 2-CNF

F(n, m) : formulae with n variables and m clauses. Critical range: m = n(1 + µn−1/3) P

• F(n, m) is SAT
• ∼

     1 − Θ( 1

|µ|3 ),

µ → −∞; Θ(1), µ = Θ(1); exp(−Θ(µ3)), 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 an,m be the #of graphs from a family F with n vertices and m edges. The exponential generating function of F is defined as F(z, w) :=

• n,m0

an,m znwm n! . 4 1 3 2 w5 z4 4! Theorem ♥. The EGF S(z, w) for strongly connected directed graphs and the EGF G(z, w) for simple graphs satisfy the relation S(z, w) = − log

• G(z, w) ⊙z

1 G(z, w)

• where ⊙z is the exponential Hadamard product
• n

an(w)zn n! ⊙z

• n

bn(w)zn n! =

• n

an(w)bn(w)zn n!

.

The results

The results

Subcritical phase: m = n(1 + µn−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 − a1 |µ|3 + a2 |µ|6 − · · ·

• Theorem. The number of contradictory

variables follows the Gamma(2) law with density f (x) ∼ xe−µn−1/3x.

• 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 1 2 3 3 1 + 2 1 2 3 3 1 = 1 2 3 1 2 3

Rotate (x1 ∨ x3)

2 1 2 3 3 1 + 2 1 2 3 3 1 = 1 2 3 1 2 3

Cherry-picking the sum-representation

1 2 3 1 3 2 5 4 5 4 = 1 2 3 1 5 4 3 2 5 4 + 1 3 2 1 5 4 2 3 5 4

P( sum-representation contains the

contradictory circuit of length ℓ ) = 2−ℓ

Typical contradictory components

Subcritical phase, m = n(1 + µn−1/3), µ → −∞

• Lemma. The contribution of a contra-

dictory cubic component with excess 2r is Θ(|µ|−3r). 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 + µn−1/3), µ → −∞

Theorem.

◮ There are 1 2n2/3µ−2 vertices belonging to the tree-like spine

components.

◮ Further contributions are c1n2/3µ−5, c2n2/3µ−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 + µn−1/3), µ = constant

Let ξ be a discrete random variable ξ ∈ {0, 1, 2, 3, · · · }. Then, P(ξ = 0) = 1 − Eξ + E ξ(ξ−1)

2!

− E ξ(ξ−1)(ξ−2)

3!

+ · · · The same event A can be expressed as [ξ = 0] for different possible ξ.

◮ P(F(n, m) is SAT) = P(ξ1 = 0) = P(ξ2 = 0) ◮ ξ1 = number of contradictory variables ◮ ξ2 = number of contradictory components

Challenge: multiple counting of the overlapping contradictory components gives divergent series for Eξ2 → +∞ when µ µc, µc < 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

T h a n k y

• u

!