Scale-Free Random SAT Instances

We focus on the random generation of SAT instances that have properties similar to real-world instances. It is known that many industrial instances, even with a great number of variables, can be solved by a clever solver in a reasonable amount of time. This is not possible, in general, with classical randomly generated instances. We provide a different generation model of SAT instances, called \emph{scale-free random SAT instances}. It is based on the use of a non-uniform probability distribution $P(i)\sim i^{-\beta}$ to select variable $i$, where $\beta$ is a parameter of the model. This results into formulas where the number of occurrences $k$ of variables follows a power-law distribution $P(k)\sim k^{-\delta}$ where $\delta = 1 + 1/\beta$. This property has been observed in most real-world SAT instances. For $\beta=0$, our model extends classical random SAT instances. We prove the existence of a SAT-UNSAT phase transition phenomenon for scale-free random 2-SAT instances with $\beta<1/2$ when the clause/variable ratio is $m/n=\frac{1-2\beta}{(1-\beta)^2}$. We also prove that scale-free random k-SAT instances are unsatisfiable with high probability when the number of clauses exceeds $\omega(n^{(1-\beta)k})$. %This implies that the SAT/UNSAT phase transition phenomena vanishes when $\beta>1-1/k$, and formulas are unsatisfiable due to a small core of clauses. The proof of this result suggests that, when $\beta>1-1/k$, the unsatisfiability of most formulas may be due to small cores of clauses. Finally, we show how this model will allow us to generate random instances similar to industrial instances, of interest for testing purposes

The Galactic Disk Distribution of Planetary Nebulae with Warm Dust Emission Features: II

Can the distribution of warm-dust compositions in IR-bright galactic disk PNe be linked to the underlying stellar population? The PNe with warm dust emission represent a homogeneous population, which is presumably young and minimally affected by a possible dependence of PN lifetime on progenitor mass. The sample in paper I thus allows testing the predictions of single star evolution, through a comparison with synthetic distributions and under the assumption that tip-of-the-AGB and PN statistics are similar. We construct a schematic model for AGB evolution (adapted from Groenewegen & de Jong 1993), whose free-parameters are calibrated with the luminosity function (LF) of C stars in the LMC, the initial-final mass relation, and the range of PN compositions. The observed metallicity gradient and distribution of star forming regions with galactocentric radius (Bronfman et al. 2000) allow us to synthesise the galactic disk PN progenitor population. We find the fraction of O-rich PNe, f(O), is a tight constraint on AGB parameters. For our best model, a minimum PN progenitor mass Mmin=1Msun predicts that about 50% of all young PNe should be O-rich, compared to an observed fraction of 22%; thus Mmin=1.2Msun, at a 2sigma confidence level. By contrast, current AGB models for single stars can account neither for the continuous range of N enrichment (Leisy & Dennefeld 1996), nor for the observation that the majority of very C-rich PNe have Peimbert type I (paper I). f(O) is thus an observable much easier to model. The decrease in f(O) with galactocentric radius, as reported in paper I, is a strong property of the synthetic distribution, independent of Mmin. This trend reflects the sensitivity of the surface temperature of AGB stars and of the core mass at the first thermal pulse to the galactic metallicity gradient.Comment: accepted by MNRA