Fig 1.
The figure shows the conflict graph generated in our example run of CDCL when solving formula (1). The decision literals are marked in gray. The conflict literals are marked in red. The reason(s) for a propagation is given as label(s) of the edge(s). The first unique implication point (1UIP) is shown in blue. The 1UIP cut is the dashed blue line leading to the learned clause . The graphic is adapted from [30].
Table 1.
Summary of the hardware used in our experiments.
Table 2.
The effect of learned clauses (without deletion) on the runtime of Glucose 4.1.
Table 3.
The effect of learned clauses (without deletion) on the runtime of Glucose 4.1 with respect to the satisfiability of the base instance.
Table 4.
The effect of learned clauses (without deletion) on the number of conflicts used by Glucose 4.1.
Table 5.
Comparison between the effect of adding clauses (without deletion) for time and number of conflicts of Glucose 4.1.
Fig 2.
Multimodal histogram of runtime distribution.
We used the Kaplan–Meier estimate to obtain the histogram of the runtime distribution of the instance UNSAT_ME_seq-sat_Thoughtful_p11_6_59-typed.pddl_43. We used the Expectation–maximization (EM) method to obtain the pdf of the fitted Weibull mixture model (see Definitions 6.1 and 6.2 for an introduction to this kind of distribution). The EM algorithm is an algorithm that allows cluster analysis by starting with a heuristically initialized model and alternating between two steps. First, in the expectation-step (E-step), the association of the data points to the different clusters gets changed. Then, in the maximization-step (M-step), the model’s parameters get improved by using this new association of the data points. We refer to the classic paper [51] for an introduction to the algorithm. The resulting fitted distribution that is seen in the plot is clearly multimodal. This is supported by a Hartigans’ dip test value of 0.015 > 0.005.
Fig 3.
Histogram of runtimes vs. histogram of logarithmically scaled runtimes.
Scaling the x-axis of a histogram logarithmically can often uncover multimodality that is not clearly visible in the unscaled histogram. The graphic depicts both histograms for the instance size_5_5_5_i019_r12, where this difference is very pronounced. (above) Histogram of CPU times with a Hartigans’ dip test value of 0.005. (below) Histogram of logarithmically scaled CPU times with a Hartigans’ dip test value of 0.016.
Fig 4.
Multimodal histogram of the distribution for the number of propagations and decisions.
The histograms show the distribution of the two measures propagations and decisions required to solve the extended instances of UNSAT_ME_seq-sat_Thoughtful_p11_6_59-typed.pddl_43. Both histograms for the hardware-independent measures have the same multimodal form as the histogram for CPU time shown in Fig 2. (above) Histogram for number of propagations with a Hartigans’ dip test value of 0.005. (below) Histogram for number of decisions with a Hartigans’ dip test value of 0.004.
Fig 5.
Mixture distribution and components.
The figure shows the Weibull components (in black) underlying the mixture distribution (in red) of instance UNSAT_ME_seq-sat_Thoughtful_p11_6_59-typed.pddl_43. The Weibull components were scaled according to their respective pi-values. For the histogram of this instance, refer to Fig 2.
Fig 6.
The Q–Q plot for instance crafted_n11_d6_c4_num19 was obtained by the quantiles of a fitted 3-parameter Weibull distribution and the data quantiles. The plot appears as a straight line. The correlation coefficient calculates to 0.9997979. For reference, the identity is given in gray.
Fig 7.
Inspection of the smallest and largest component of the Weibull mixture model.
Based on the Kaplan–Meier estimator, the estimations of the cdf and survival function of the multimodal instance bivium-40–200 are shown (Hartigans’ dip test value 0.010). Both the left and the right tails appear as straight lines (depicted in red). The plot of the cdf is a log–log plot, while the plot of the survival function is a loglog–log plot. The gray area marks the confidence interval. This suggests that the smallest and largest component of the underlying mixture model are Weibull distributions. (left) Estimation of the cdf. (right) Estimation of the survival function.
Fig 8.
This figure shows various plots of the unimodal instance 6g_5color_164_100_01 (Hartigans’ dip test value 0.003). This is an example of an instance with a long-tailed runtime distribution. (a) The plot shows the histogram of runtimes (in gray) and the fitted pdf (in red). Both are shifted to the left by the minimal time T0 required to solve any extended instance. The obtained shape parameter of the fit is k = 0.884 < 1. Thus, the distribution is long-tailed. (b) We have plotted the logarithm of the tail of the distribution, i. e., logS(x). By visual inspection, one can see that it decays sub-linearly. In this case, . This property characterizes the class of so-called heavy-tailed distributions (a superset of the class of long-tailed distributions) [55]. Intuitively, this means that the algorithm has a non-vanishing probability of requiring very long runtimes. For comparison, we have plotted the logarithmic survival function of an exponentially distributed random variable with the same expectation in blue. The logarithm of the tail of such an exponential distribution decays linearly. (c) Zoomed in version of (b). This clearly shows the sublinear decay by focusing on the curvature.
Fig 9.
Multimodal and long-tailed effect with MiniSAT.
The histogram shows the distributions of CPU times when solving the extended instances of 6g_5color_164_100_01 with MiniSAT. Both a multimodal behavior as well as a long-tailed effect are visible. The multimodality is confirmed by a Hartigans’ dip test value of 0.031.