(a) Stingy SAT is the following problem: given a set of clauses (each a disjunction of literals) and an integer k, find a satisfying assignment in which at most k variables are true, if such an assignment exists. Prove that stingy SAT is NP-hard. (b) The Double SAT problem asks whether a given satisfiability problem has at least two different satisfying assignments. For example, the problem {{V1, V2}, {V1, V2}, {V1, V2}} is satisfiable, but has only one solution (v₁ = F, v₂ = T). In contrast, {{V1, V2}, {V1, V2}} has exactly two solutions. Show that Double-SAT is NP-hard.
(a) Stingy SAT is the following problem: given a set of clauses (each a disjunction of literals) and an integer k, find a satisfying assignment in which at most k variables are true, if such an assignment exists. Prove that stingy SAT is NP-hard. (b) The Double SAT problem asks whether a given satisfiability problem has at least two different satisfying assignments. For example, the problem {{V1, V2}, {V1, V2}, {V1, V2}} is satisfiable, but has only one solution (v₁ = F, v₂ = T). In contrast, {{V1, V2}, {V1, V2}} has exactly two solutions. Show that Double-SAT is NP-hard.
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![(a) Stingy SAT is the following problem: given a set of clauses (each a disjunction of
literals) and an integer k, find a satisfying assignment in which at most k variables
are true, if such an assignment exists. Prove that stingy SAT is NP-hard.
(b) The Double SAT problem asks whether a given satisfiability problem has at least two
different satisfying assignments. For example, the problem {{V1, V2}, {V1, V2}, {V1, V2}}
is satisfiable, but has only one solution (v₁ = F, v₂ = T). In contrast, {{V1, V2}, {V1, V2}}
has exactly two solutions. Show that Double-SAT is NP-hard.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F840ac9d4-eb27-4280-b70d-21eed9a81895%2F51cb33f4-ca88-47ae-a3b2-4ce015c3f14a%2F87ngrw_processed.png&w=3840&q=75)
Transcribed Image Text:(a) Stingy SAT is the following problem: given a set of clauses (each a disjunction of
literals) and an integer k, find a satisfying assignment in which at most k variables
are true, if such an assignment exists. Prove that stingy SAT is NP-hard.
(b) The Double SAT problem asks whether a given satisfiability problem has at least two
different satisfying assignments. For example, the problem {{V1, V2}, {V1, V2}, {V1, V2}}
is satisfiable, but has only one solution (v₁ = F, v₂ = T). In contrast, {{V1, V2}, {V1, V2}}
has exactly two solutions. Show that Double-SAT is NP-hard.
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