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SAT stands for the Boolean satisfiability problem.
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Does 1-in-3 SAT remain NP-hard even if every variable occurs both positively and negatively?
The standard problem 1-in-3 SAT (or XSAT or X3SAT) is:
Instance: a CNF formula with every clause containing exactly 3 literals
Question: is there a satisfying assignment setting precisely 1 literal per … The usual reduction from 3SAT showing that 1-in-3 SAT is hard replaces a clause $(x\lor y \lor z)$ by clauses $(\lnot x \lor a \lor b)$, $(y\lor b\lor c)$, $(\lnot z \lor c \lor d)$ where $a,b,c,d$ are …