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Inhomogeneous random 2-SAT

Jan Hladký, Petr Savický

TL;DR

This work generalizes random 2-SAT by introducing inhomogeneity through a graphon/digraphon kernel W and random type assignments to variables. The authors connect satisfiability to spectral properties of an associated integral operator, establishing a sharp threshold at ρ^*(W)=1: subcritical regimes (ρ^*(W)<1) yield a.a.s. satisfiable formulas, while supercritical regimes (ρ^*(W)>1) yield a.a.s. unsatisfiable formulas. The analysis combines a digraph-onnections approach via implication digraphons, a detailed bicycle/snakes-based second-moment framework, and advanced Banach-space spectral theory for non-continuous kernels. The results naturally extend the classical homogeneous 2-SAT threshold and offer a versatile toolkit for inhomogeneous and scale-free random formulas, with potential extensions to higher clause sizes. The work lays foundational theory for graphon-based satisfiability and highlights rich connections to stochastic block models, spectral graph theory, and digraphon analysis.

Abstract

We introduce an inhomogeneous variant of random 2-SAT. Each variable $v_1,\ldots,v_n$ is assigned a type from a state space $Λ$, independently at random. Clause inclusion is governed by a symmetric measurable kernel $W$ on $(Λ\times \{+,-\})^2$, in analogy with the inhomogeneous random graph model of Bollobás, Janson, and Riordan: given literals $\ell_i\in\{v_i,\neg v_i\}$ and $\ell_j\in\{v_j,\neg v_j\}$, the clause $\{\ell_i,\ell_j\}$ appears with probability $W(\mathrm{type}(\ell_i),\mathrm{type}(\ell_j))/(2n)$. In particular, for a variable $v_i$ of type $x\inΛ$, the slices $W((+,x),\cdot)$ and $W((-,x),\cdot)$ describe how $v_i$ and $\neg v_i$ interact with other literals. We identify a parameter $ρ^*(W)$, defined as the spectral radius of an integral operator derived from $W$, and show that $ρ^*(W)<1$ and $ρ^*(W)>1$ correspond to asymptotically almost surely satisfiable and unsatisfiable instances, respectively. The satisfiability threshold for homogeneous random 2-SAT is well-established, occurring when the ratio of clauses to variables is $1$. This corresponds to a weight function of $W \equiv 1$ and a clause density of $1/(2n)$. Our result extends this classical result to a broad class of models controlled by types of variables.

Inhomogeneous random 2-SAT

TL;DR

This work generalizes random 2-SAT by introducing inhomogeneity through a graphon/digraphon kernel W and random type assignments to variables. The authors connect satisfiability to spectral properties of an associated integral operator, establishing a sharp threshold at ρ^*(W)=1: subcritical regimes (ρ^*(W)<1) yield a.a.s. satisfiable formulas, while supercritical regimes (ρ^*(W)>1) yield a.a.s. unsatisfiable formulas. The analysis combines a digraph-onnections approach via implication digraphons, a detailed bicycle/snakes-based second-moment framework, and advanced Banach-space spectral theory for non-continuous kernels. The results naturally extend the classical homogeneous 2-SAT threshold and offer a versatile toolkit for inhomogeneous and scale-free random formulas, with potential extensions to higher clause sizes. The work lays foundational theory for graphon-based satisfiability and highlights rich connections to stochastic block models, spectral graph theory, and digraphon analysis.

Abstract

We introduce an inhomogeneous variant of random 2-SAT. Each variable is assigned a type from a state space , independently at random. Clause inclusion is governed by a symmetric measurable kernel on , in analogy with the inhomogeneous random graph model of Bollobás, Janson, and Riordan: given literals and , the clause appears with probability . In particular, for a variable of type , the slices and describe how and interact with other literals. We identify a parameter , defined as the spectral radius of an integral operator derived from , and show that and correspond to asymptotically almost surely satisfiable and unsatisfiable instances, respectively. The satisfiability threshold for homogeneous random 2-SAT is well-established, occurring when the ratio of clauses to variables is . This corresponds to a weight function of and a clause density of . Our result extends this classical result to a broad class of models controlled by types of variables.
Paper Structure (38 sections, 32 theorems, 92 equations, 4 figures)

This paper contains 38 sections, 32 theorems, 92 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our model $\mathsf{TwoSAT}(n,W)$
  3. Stochastic block model
  4. The statement of the result
  5. Theorem \ref{['Inhomo2SAT:thm:main']} and scale-free random formulea
  6. Alternative scalings and graphons with $\rho^*(W)=0$
  7. Solving 2-SAT using the implication digraph
  8. A convenient modification of $\mathsf{TwoSAT}(n,W)$
  9. Random digraph $\mathbb{G}(n,\overrightarrow{W})$
  10. Basic idea behind the proof of the main theorem
  11. Comparison to previous work on inhomogeneous random (di)graphs
  12. Preliminaries
  13. Measure theory
  14. Digraphons
  15. Banach space theory
  16. ...and 23 more sections

Key Result

Theorem 1.4

Suppose that $\Gamma$ is an $L^1$-digraphon on $\Omega$. Then there exists a finite or a countable set $I$ not containing $0$ and a decomposition $\Omega=X_0\sqcup \bigsqcup_{i\in I} X_i$ so that $X_0$ is either an empty set or is fragmented in $\Gamma$ and each $X_i$ is a strong component. Further,

Figures (4)

  • Figure 1: A visualization of the transformation of a graphon $W$ into its implication digraphon $\overrightarrow{W}$. The graphon $W$ consists of four parts: a symmetric part $A\in L^1\left((\Lambda\times\{+\})\times (\Lambda\times\{+\})\right)$, a symmetric part $C\in L^1\left((\Lambda\times\{-\})\times (\Lambda\times\{-\})\right)$ and a pair of mutually transposed parts $B$ and $B^T$, where $B\in L^1\left((\Lambda\times\{+\})\times (\Lambda\times\{-\})\right)$, $B^T\in L^1\left((\Lambda\times\{-\})\times (\Lambda\times\{+\})\right)$. (The orientation of the plane here and elsewhere follows the matrix convention, that is, the main diagonal is in the $\searrow$ direction.)
  • Figure 2: A formula $\phi=(\neg v_1 \vee v_2) \wedge (\neg v_2 \vee v_3) \wedge (\neg v_3 \vee v_1) \wedge (v_1 \vee v_2)\wedge (\neg v_3 \vee \neg v_2)$, and its implication digraph $\mathsf{D}(\phi)$. One contradictory cycle is highlighted.
  • Figure 3: An example of a serpent when $a=6$, $b=4$. The serpent is depicted in red. Its free literals are depicted in full red circles, its non-free literals are depicted in half-red circles. Its intersection edges are dotted. The good non-intersection sequences written in the format $(start,end,length)$ are: $(g,l_5,2)$, $(\neg l_8,\neg g,2)$, $(\neg g,l_7,1)$.
  • Figure 4: An example of two serpents which do not have the same intersection pattern but have the same set of good non-intersection sequences. Written in the format $(start,end,length)$, these are: $(g,l_6,1)$, $(l_1,\neg g,1)$, $(\neg g,g,3)$. Here, $a=7$, $b=3$.

Theorems & Definitions (63)

  • Definition 1.1: implication digraphon
  • Definition 1.2: restriction
  • Definition 1.3: strongly connected set, strong component, fragmented set
  • Theorem 1.4: Theorem \ref{['DIGRAPHONS-thm:decompositionIntoComponents']} in HladkySavicky:Digraphons
  • Definition 1.5: contradictory set
  • Definition 1.6: eigenvalues, eigenfunctions, spectrum, point spectrum, spectral radius
  • Remark 1.7
  • Definition 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 53 more