Inhomogeneous random 2-SAT
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866908655742353408 |
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| author | Hladký, Jan Savický, Petr |
| author_facet | Hladký, Jan Savický, Petr |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_17656 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Inhomogeneous random 2-SAT Hladký, Jan Savický, Petr Combinatorics Probability 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. |
| title | Inhomogeneous random 2-SAT |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2510.17656 |