In computational complexity theory, a decision problem is NP-complete when it is both in NP and NP-hard. Although a solution to a NP-complete can be verified quickly, there is no known algorithm to solve it in polynomial time. There exists a method to reduce a SAT (Satifiability) problem to Subset Sum Problem (SSP) in the literature, however, it can only be applied to small or medium size problems. Our study is to find an efficient method to transform a SAT problem to a mixed integer linear programming problem in larger size. Observing the feature of variable-clauses constraints in SAT, we apply linear inequality model (LIM) to the problem and propose a method called LIMSAT. The new method can work efficiently for very large size problem with thousands of variables and clauses in SAT tested using up-to-date benchmarks.
Gilles @cite_13 study how to measure SAT instances. They give 5 indicators: the number of decision levels, the number of decisions between two conflicts, the number of successive conflicts, the number of non-binary glue clauses and the number of unit propagation. They also mention the restart polarity policy which is added to Glucose. The new version of Glucose solves 20$
@cite_13
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With the increasing performance of SAT solvers, a lot of distinct problems, coming from very disparate fields, are added to the pool of Application problems, regularly used to rank solvers. These problems are also widely used to measure the positive impact of any new idea. We show in this paper that many of them have extreme behaviors that any SAT solvers must cope with. We show that, by adding a few, simple, human-readable, indicators, we can let Glucose choose between four strategies to show important improvements on the set of the hardest problems from all the competitions between 2002 and 2013 included. Moreover, once the SAT solver has been specialized, we show that a new restart polarity policy can improve even more the results. Without the first specialization step mentioned above, this new and effective policy would have been jugged inefficient. Our final Glucose is capable of solving (20 , ) more problems than the original one, while speeding up also UNSAT answers. We present an exact algorithm that decides, for every fixed r ≥ 2 in time O(m) + 2O(k2) whether a given set of m clauses of size r admits a truth assignment that satisfies at least ((2r - 1)m + k) 2r clauses. Thus Max-r-Sat is fixed-parameter tractable when parameterized by the number of satisfied clauses above the tight lower bound (1--2-r)m. This solves an open problem of Mahajan, Raman and Sikdar (J. Comput. System Sci., 75, 2009). Our algorithm is based on a polynomial-time data reduction procedure that reduces a problem instance to an equivalent algebraically represented problem with O(k2) variables. This is done by representing the instance as an appropriate polynomial, and by applying a probabilistic argument combined with some simple tools from Harmonic analysis to show that if the polynomial cannot be reduced to one of size O(k2), then there is a truth assignment satisfying the required number of clauses. Combining another probabilistic argument with tools from graph matching theory and signed graphs, we show that if an instance of Max-2-Sat with m clauses has at least 3k variables after application of certain polynomial time reduction rules to it, then there is a truth assignment that satisfies at least (3m + k) 4 clauses. We also outline how the fixed-parameter tractability result on Max-r-Sat can be extended to a family of Boolean Constraint Satisfaction Problems. In this paper, we present a divide-and-conquer SAT solver, MapleAmpharos, that uses a novel propagation-rate (PR) based splitting heuristic. The key idea is that we rank variables based on the ratio of how many propagations they cause during the run of the worker conflict-driven clause-learning solvers to the number of times they are branched on, with the variable that causes the most propagations ranked first. The intuition here is that, in the context of divide-and-conquer solvers, it is most profitable to split on variables that maximize the propagation rate. Our implementation MapleAmpharos uses the AMPHAROS solver as its base. We performed extensive evaluation of MapleAmpharos against other competitive parallel solvers such as Treengeling, Plingeling, Parallel CryptoMiniSat5, and Glucose-Syrup. We show that on the SAT 2016 competition Application benchmark and a set of cryptographic instances, our solver MapleAmpharos is competitive with respect to these top parallel solvers. What is surprising that we obtain this result primarily by modifying the splitting heuristic. Many NP-complete constraint satisfaction problems appear to undergo a “phase transition” from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment (expectation) of the number of solutions tends to zero. We show that in the case of certain symmetric constraints, considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold. Specifically, we prove that the threshold for both random hypergraph 2-colorability (Property B) and random Not-All-Equal @math -SAT is @math . As a corollary, we establish that the threshold for random @math -SAT is of order @math , resolving a long-standing open problem.