Exact Satisfiabitity with Jokers

The XSAT problem asks for solutions of a set of clauses where for every clause exactly one literal is satisfied. The present work investigates a variant of this well-investigated topic where variables can take a joker-value (which is preserved by negation) and a clause is satisfied iff either exactly one literal is true and no literal has a joker value or exactly one literal has a joker value and no literal is true. While JX2SAT is in polynomial time, the problem becomes NP-hard when one searches for a solution with the minimum number of jokers used and the decision problem X3SAT can be reduced to the decision problem of the JX2SAT problem with a bound on the number of jokers used. JX3SAT is in both cases, with or without optimisation of the number of jokers, NP-hard and X3SAT can be reduced to it without increasing the number of variables. Furthermore, the general JXSAT problem can be solved in the same amount of time as variable-weighted XSAT and the obtained solution has the minimum amount of number of jokers possible.