Exact enumeration of satisfiable 2-SAT formulae

Dovgal, Sergey; Panafieu, Élie de; Ravelomanana, Vlady

doi:10.48550/arxiv.2108.08067

Cited by 1 publication

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“…The current article aims to systematize and solidify the methods presented there. Afterwards, the method has been extended even further to the case of directed hypergraphs [59] and enumeration of satisfiable 2-SAT formulae [16]. In Section 3.1, we recall the definition of the arrow product, a key combinatorial operation on digraph families, and its connection with graphic generating functions.…”

Section: Symbolic Methods For Directed Graphsmentioning

confidence: 99%

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The birth of the strong components

Panafieu

Dovgal

Ralaivaosaona

et al. 2023

Random Struct Algorithms

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It is known that random directed graphs undergo a phase transition around the point . Earlier, Łuczak and Seierstad have established that as when , the asymptotic probability that the strongly connected components of a random directed graph are only cycles and single vertices decreases from 1 to 0 as goes from to . By using techniques from analytic combinatorics, we establish the exact limiting value of this probability as a function of and provide more statistical insights into the structure of a random digraph around, below and above its transition point. We obtain the limiting probability that a random digraph is acyclic and the probability that it has one strongly connected complex component with a given difference between the number of edges and vertices (called excess). Our result can be extended to the case of several complex components with given excesses as well in the whole range of sparse digraphs. Our study is based on a general symbolic method which can deal with a great variety of possible digraph families, and a version of the saddle point method which can be systematically applied to the complex contour integrals appearing from the symbolic method. While the technically easiest model is the model of random multidigraphs, in which multiple edges are allowed, and where edge multiplicities are sampled independently according to a Poisson distribution with a fixed parameter , we also show how to systematically approach the family of simple digraphs, where multiple edges are forbidden, and where 2‐cycles are either allowed or not. Our theoretical predictions are supported by numerical simulations when the number of vertices is finite, and we provide tables of numerical values for the integrals of Airy functions that appear in this study.

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Section: Symbolic Methods For Directed Graphsmentioning

confidence: 99%

“…FIGURE16 Numerical plot of the theoretical limit of the probability P(n, p) that a random multidigraph has one bicyclic complex component inside the critical window for 𝜇 ∈ [−5, 3].…”

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confidence: 99%