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Great Ideas in Computing Complexity Theory

Great Ideas in Computing Complexity Theory. Richard Anderson University of Washington. Mathematical Structure of Computation. What is computation? Church’s Thesis – all forms of computation are equivalent. Alan Turing. Undecidability of the halting problem

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Great Ideas in Computing Complexity Theory

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  1. Great Ideas in ComputingComplexity Theory Richard Anderson University of Washington IUCEE: Discrete Mathematics

  2. Mathematical Structure of Computation • What is computation? • Church’s Thesis – all forms of computation are equivalent IUCEE: Discrete Mathematics

  3. Alan Turing • Undecidability of the halting problem • A function that provably cannot be computed IUCEE: Discrete Mathematics

  4. Combinatorial Optimization • Ford-Fulkerson 1956 IUCEE: Discrete Mathematics

  5. Jack Edmonds • Polynomial Time IUCEE: Discrete Mathematics

  6. Non deterministic Turing Machine IUCEE: Discrete Mathematics

  7. Hard problems • Satisfiability • Bin Packing • Integer Programming • Hamiltonian Circuit • Vertex Cover • 3 Dimensional Matching • Traveling Salesman Problem IUCEE: Discrete Mathematics

  8. NP Completeness • Non-deterministic polynomial time • Cook’s theorem: • Satisfiability is the hardest problem in NP • Simulate a polynomial time non-deterministic computation with satisfiability formula • Karp • Showed that a wide range of other problems were also NP-complete • Showed how to convert satisfiability into TSP IUCEE: Discrete Mathematics

  9. Satisfiability • Given a boolean formula, is there an assignment of the variables to make it true • Simplified version • CNF • Each clause has at most 3 literals (x || y || z) && (!x || !y || !z) && (!x || y) && (x || !y) && (y || !z) && (!y || z) IUCEE: Discrete Mathematics

  10. Simulation of a formula with a path in a graph • G has a Hamiltonian Circuit if and only if F has a satisfying truth assignment • G can be constructed “easily” from F IUCEE: Discrete Mathematics

  11. Gadgets: Truth Setting IUCEE: Discrete Mathematics

  12. Gadgets: Truth Testing IUCEE: Discrete Mathematics

  13. Papadimitriou • Hamiltonian Circuit NP Complete for Grid Graphs IUCEE: Discrete Mathematics

  14. Euclidean TSP • n points in a Rn • Distance between a pair of points is the Euclidean distance • Is the Euclidean TSP NP-complete? IUCEE: Discrete Mathematics

  15. On beyond NP NP-Complete P-SPACE NP P IUCEE: Discrete Mathematics

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