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FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY

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FORMAL LANGUAGES, AUTOMATA AND COMPUTABILITY

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    10. NP Complete Problem: hardest problem in NP Intuitively, L is harder than L’ if L’ is polynomial reducible to L. NP Complete Problem: If such a problem has an efficient algorithm (in P), then every other problem also has efficient algorithm.

    13. NP-Complete Problems 3SAT k-Clique Vertex Cover Independent Set

    14. 3SAT Problem: Given a CNF where each clause has 3 variables, decide whether it is satisfiable or not. (x1 ? x1 ? x2) ? (?x1 ? ?x2 ? ?x2) ? (?x1 ? x2 ? x2)

    17. How to prove? We can convert (in polynomial time) a given SAT instance S into a 3SAT instance S’ such that If S is satisfiable, then S’ is satisfiable. If S’ is satisfiable, then S is satisfiable.

    19. Polynomial Time Reduction Clause in SAT x1 x1 ? ? x2 x1 ? x2 ? x3 x1 ? x2 ? x3 ? x4 x1 ? x2 ? x3 ? x4 ? x5 Clauses in 3SAT x1 ? x1 ? x1 x1 ? x1 ? ? x2 x1 ? x2 ? x3 (x1 ? x2 ? z1)? ( ? z1 ? x3 ? x4) (x1 ? x2 ? z1)? ( ? z1 ? x3 ? z2)? (? z2 ? x4 ? z3)? (? z3? x4 ? x5)

    20. Finding cliques is a useful operation in social networks, computing protein similarity, finding error correcting codes, there are all kinds of applications. For example, this graph here has a 5-clique… Now it would seem that this problem may not be all that hard. After all, it certainly LOOKS easy!Finding cliques is a useful operation in social networks, computing protein similarity, finding error correcting codes, there are all kinds of applications. For example, this graph here has a 5-clique… Now it would seem that this problem may not be all that hard. After all, it certainly LOOKS easy!

    22. Why is (1) true? (1) is easy: if (G,k) in CLIQUE, then we can just give the k-clique to a verifier, and a verifier can check that all pairs of nodes in the k-clique have an edge.Why is (1) true? (1) is easy: if (G,k) in CLIQUE, then we can just give the k-clique to a verifier, and a verifier can check that all pairs of nodes in the k-clique have an edge.

    25. not contradictory: that is, one of the nodes is labeled x, and the other node is labeled not(x).not contradictory: that is, one of the nodes is labeled x, and the other node is labeled not(x).

    26. Now when we find a 3-clique, what are we doing? We’re picking out a literal in each of the clauses so that, taken as a whole, no pair of literals are contradictory. This corresponds to a satisfying assignment!Now when we find a 3-clique, what are we doing? We’re picking out a literal in each of the clauses so that, taken as a whole, no pair of literals are contradictory. This corresponds to a satisfying assignment!

    29. Independent Set

    31. Key Observation For a graph G, vertex set S is an independent set if and only if S is clique in G*.

    33. In the vertex cover problem, we want to find a set of nodes such that every edge is “covered” by the nodes. That is, at least one endpoint of the edge is in the set. What’s a vertex cover for this graph? (The set of all nodes!) What’s a minimum size vertex cover? In the vertex cover problem, we want to find a set of nodes such that every edge is “covered” by the nodes. That is, at least one endpoint of the edge is in the set. What’s a vertex cover for this graph? (The set of all nodes!) What’s a minimum size vertex cover?

    34. Vertex Cover is NP Complete Given a Graph G(V,E), decide if there is k vertex such that every edge is covered by one of them?

    35. K-Indep Set ?P Vertex Cover

    36. Key Observation For a graph G(V,E), S is a independent set if and only if V-S is a vertex cover.

    37. Other NP-Complete Problems Travelling Salesman Problem Hamiltonian Path Max Cut Subset Sum Integer Programming ….

    38. Other Problems in NP Graph Isomorphism Factoring Number We don’t know if they are NP Complete or not.

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