# 2SAT - WhereCanWeDrawTheLine? OntheHardnessofSatisfiability...

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1 Where Can We Draw The Line? On the Hardness of Satisfiability  Problems

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2 Introduction Objectives: To show variants of  SAT  and check if they are  NP-hard   Overview: Known results 2SAT Max2SAT
3 What Do We Know? Checking if a propositional calculus formula is  satisfiable ( SAT ) is  NP-hard . ¬ (x ∧¬ z ( ¬ w x)) (x y) →¬ y ¬ (x z ( ¬ w x)) (x y) y Example: propositional calculus formula

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4 What Do We Know? We concentrated on a special case:  CNF  formulas.    (.. .. ..... ..) (.. .. ..... ..)  (.. .. ..... ..) (.. .. ..... ..)  structure of CNF formulas
5 What Do We Know? maximal number of  literals per clause 1 2 3 4 P NP-hard We’ll explore  this!

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6 2SAT Instance:  A  2-CNF  formula  ϕ Problem:  To decide if  ϕ  is satisfiable ( ¬ x y) ( ¬ y z) (x ∨¬ z) (z y) ( ¬ x y) ( ¬ y z) (x z) (z y) Example: a 2CNF formula
7 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… PAP 184-185

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8 Searching in Graphs Theorem:  Given a graph  G=(V,E)  and two vertices  s,t V , finding if there is a path from  s  to  t  in  G  is  polynomial-time decidable.
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## This note was uploaded on 10/23/2011 for the course CS 4540 taught by Professor Staff during the Fall '10 term at Georgia Institute of Technology.

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2SAT - WhereCanWeDrawTheLine? OntheHardnessofSatisfiability...

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