On variable-weighted exact satisfiability problems

We show that the NP-hard optimization problems minimum and maximum weight exact satisfiability (XSAT) for a CNF formula C over n propositional variables equipped with arbitrary real-valued weights can be solved in O(|C|2^{0.2441n}) time. To the best of our knowledge, the algorithms presented here are the first handling weighted XSAT optimization versions in non-trivial worst case time. We also investigate the corresponding weighted counting problems, namely we show that the number of all minimum, resp. maximum, weight exact satisfiability solutions of an arbitrarily weighted formula can be determined in O(n^2cdot |C|+2^{0.40567n}) time. In recent years only the unweighted counterparts of these problems have been studied cite{dahl,dahl2,porschen}

On variable-weighted exact satisfiability problems

We show that the NP-hard optimization problems minimum and maximum weight exact satisfiability (XSAT) for a CNF formula C over n propositional variables equipped with arbitrary real-valued weights can be solved in O(|C|2^{0.2441n}) time. To the best of our knowledge, the algorithms presented here are the first handling weighted XSAT optimization versions in non-trivial worst case time. We also investigate the corresponding weighted counting problems, namely we show that the number of all minimum, resp. maximum, weight exact satisfiability solutions of an arbitrarily weighted formula can be determined in O(n^2cdot |C|+2^{0.40567n}) time. In recent years only the unweighted counterparts of these problems have been studied cite{dahl,dahl2,porschen}

Algorithms for Variable-Weighted 2-SAT and Dual Problems

In this paper we study NP-hard weighted satisfiability optimization problems for the class 2-CNF providing worst-case upper time bounds. Moreover we consider the monotone dual class consisting of clause sets where all variables occur at most twice. We show that weighted SAT, XSAT and NAESAT optimization problems for this class are polynomial time solvable using appropriate reductions to specific polynomial time solvable graph problems

Algorithms for Variable-Weighted 2-SAT and Dual Problems

In this paper we study NP-hard weighted satisfiability optimization problems for the class 2-CNF providing worst-case upper time bounds. Moreover we consider the monotone dual class consisting of clause sets where all variables occur at most twice. We show that weighted SAT, XSAT and NAESAT optimization problems for this class are polynomial time solvable using appropriate reductions to specific polynomial time solvable graph problems

Solving the Cubic Monotone 1-in-3 SAT Problem in Polynomial Time

The exact 3-satisfiability problem X3SAT is known to remain NP-complete when restricted to expressions where every variable has exactly three occurrences even in the absence of negated variables Cubic Monotone 1-in-3 SAT Problem The present paper shows that the Cubic Monotone 1-in-3 SAT Problem can be solved in polynomial time and therefore prove that the conjecture P NP hold