SAT_KSAT_Safra

SAT_KSAT_Safra - Where Can We Draw The Line? On the...

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Complexity 1 Where Can We Draw The Line? On the Hardness of Satisfiability Problems
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Complexity 2 Introduction • Objectives: – To show variants of SAT and check if they are NP-hard •O v e r v i ew : – Known results –2SAT –Max2SAT
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Complexity 3 What Do We Know? • Checking if a propositional calculus formula is satisfiable ( SAT ) is NP- hard . Example: propositional calculus formula ¬ (x ∧¬ z ( ¬ w x)) (x ∧¬ y) →¬ y ¬ (x ∧¬ z ( ¬ w x)) (x ∧¬ y) →¬ y
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Complexity 4 What Do We Know? • We concentrated on a special case: CNF formulas. structure of CNF formulas (.. .. ..... ..) (.. .. ..... ..) (.. .. ..... ..) (.. .. ..... ..)
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Complexity 5 What Do We Know? maximal number of literals per clause P NP-hard We’ll explore this! 4 3 2 1
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Complexity 6 2SAT • Instance: A 2-CNF formula ϕ • Problem: To decide if ϕ is satisfiable Example: a 2CNF formula ( ¬ x y) ( ¬ y z) (x ∨¬ z) (z y) ( ¬ x y) ( ¬ y z) (x ∨¬ z) (z y)
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Complexity 7 PAP 184-185 2SAT is in P Theorem: 2SAT is polynomial-time decidable. Proof: We’ll show how to solve this problem efficiently using path searches in graphs…
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Complexity 8 Searching in Graphs Theorem: Given a graph G=(V,E) and two vertices s,t V , finding if there is
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SAT_KSAT_Safra - Where Can We Draw The Line? On the...

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