Positive 1-in-3-SAT admits a non-trivial kernel
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This paper illustrates the power of Gaussian Elimination by adapting it to the problem of Exact Satisfiability. For 1-in-3 SAT instances with non-negated literals we are able to obtain considerably smaller equivalent instances of 0/1 Integer Programming restricted to equality only. Thus we obtain an upper bound for the complexity of its counting version of O(2κ r 2^(1-κ)r) for number of variables r and clauses-to-variables ratio κ. Combining this method with previous known results gives a time and space complexity for the counting problem of O(4/3|V|2^3|V|/8) and O(4/3|V|2^3|V|/16). Our method shows that Positive instances of 1-in-3 SAT may be reduced to significantly smaller instances of I.P. in the following sense: any such instance of |V| variables and |C| clauses can be polynomial-time reduced to an instance of 0/1 Integer Programming with equality only, of size at most 2/3|V| variables and at most |C| clauses. We then proceed to define formally the notion of a non-trivial kernel. For this, we define the problems considered as Constraint Satisfaction Problems. Considering recent advances in Computational Complexity relating to Sparsification and existence of non-trivial kernels, we conclude by showing that the method presented here, giving a non-trivial kernel for positive 1-in-3 SAT, implies the existence of a non-trivial kernel for 1-in-3 SAT.
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