On the Average Number of Solutions for SAT Instances

In this paper, we are interested in counting solutions for instances of satisfiability and more precisely we try to extend the formula for the average number of solutions of random instances proposed in [4] to a large class of instances. In fact the formula given in this reference works for the specific class of instances where all clauses are dependent. When we consider the independence characteristic of clauses, we find a more general mathematical expression. The computation of the average number of solutions with the actual formula depends only on the structure of independence of the instance. The latter is defined to be the set of subsets of independent clauses. Searching the structure of independence for an instance is shown to be NP-hard. An algorithm in time O (nk2)  is designed for finding an approximate structure of an instance, k being the number of clauses and n the number of variables of the instance. Even though an approximate structure of independence is considered in the calculation of the average number of solutions, our formula yields results with more accuracy

Genetic algorithms and scatter search for MAX-SAT problems

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Vadose zone water stable isotope profiles for assessing groundwater recharge: Sensitivity to seasonal soil sampling

Groundwater recharge is widely recognized as being the most important parameter for the sustainable management of water resources. In semiarid environments, groundwater recharge can be quantified using the piston displacement method (PDM). From a single soil sampling campaign, the PDM relies on linking the deeper vadose zone soil pore water stable isotope composition (δ2HH2O and δ18OH2O) to the local meteoric water line (LMWL). However, the isotopic composition of precipitation changes seasonally, influencing the water isotope composition of the vadose zone over time. Thus, it is important to test whether the PDM is sensitive to seasonal soil sampling and whether the assessed recharge rate is independent of the time of sampling. This study investigates the effect of seasonal soil sampling on the distribution of vadose zone stable isotope composition to determine whether the sampling time influences the estimate of recharge rate from PDM. Soil samples were obtained along vertical profiles through the vadose zone in a semiarid region during the spring, summer, and autumn seasons. Specifically, the δ2HH2O and δ18OH2O of the soil pore water were determined along vertical profiles, and the PDM was applied to quantify the annual recharge. The δ2HH2O and δ18OH2O values range from −7.3 to −3.5 ‰ and from −54.4 to +7.41 ‰, respectively, and plot along a continuum with a slope less than the LMWL. Samples from deeper in the vadose zone profile had distinct ranges in isotopic composition between the three soil sampling campaigns, with isotopic composition of spring sampling dominated by lower values and those from autumn with higher values. Despite these differences, the resulting annual recharge rates from the different sampling campaigns are comparable (1.5 to 2 % of annual precipitation). Even though the pore water isotopic composition changed over time, the shift between the deeper vadose zone isotopic compositions and the LMWL remained relatively constant, leading to a similar recharge estimate over time. Therefore, the PDM-based recharge assessment in the tested semiarid environment is independent of the sampling time, which indicates that sampling for assessing groundwater recharge can be undertaken during any season