Mathematical Structure of Computation What is computation? Great - PDF document

Mathematical Structure of Computation • What is computation? Great Ideas in Computing Complexity Theory Richard Anderson • Church’s Thesis – all forms of computation are equivalent University of Washington July 1, 2008 IUCEE: Discrete Mathematics 1 July 1, 2008 IUCEE: Discrete Mathematics 2 Alan Turing Combinatorial Optimization • Undecidability of the halting problem • Ford-Fulkerson 1956 • A function that provably cannot be computed July 1, 2008 IUCEE: Discrete Mathematics 3 July 1, 2008 IUCEE: Discrete Mathematics 4 Non deterministic Turing Jack Edmonds Machine • Polynomial Time July 1, 2008 IUCEE: Discrete Mathematics 5 July 1, 2008 IUCEE: Discrete Mathematics 6

Hard problems NP Completeness • Satisfiability • Non-deterministic polynomial time • Bin Packing • Cook’s theorem: – Satisfiability is the hardest problem in NP • Integer Programming • Simulate a polynomial time non-deterministic • Hamiltonian Circuit computation with satisfiability formula • Vertex Cover • Karp • 3 Dimensional Matching – Showed that a wide range of other problems were also NP-complete • Traveling Salesman Problem • Showed how to convert satisfiability into TSP July 1, 2008 IUCEE: Discrete Mathematics 7 July 1, 2008 IUCEE: Discrete Mathematics 8 Simulation of a formula with a Satisfiability path in a graph • Given a boolean formula, is there an • G has a Hamiltonian Circuit if and only if F assignment of the variables to make it true has a satisfying truth assignment • Simplified version • G can be constructed “easily” from F – CNF – Each clause has at most 3 literals (x || y || z) && (!x || !y || !z) && (!x || y) && (x || !y) && (y || !z) && (!y || z) July 1, 2008 IUCEE: Discrete Mathematics 9 July 1, 2008 IUCEE: Discrete Mathematics 10 Gadgets: Truth Setting Gadgets: Truth Testing 12 July 1, 2008 IUCEE: Discrete Mathematics 11 July 1, 2008 IUCEE: Discrete Mathematics

Papadimitriou Euclidean TSP • n points in a R n • Hamiltonian Circuit NP Complete for Grid Graphs • Distance between a pair of points is the Euclidean distance • Is the Euclidean TSP NP-complete? July 1, 2008 IUCEE: Discrete Mathematics 13 July 1, 2008 IUCEE: Discrete Mathematics 14 On beyond NP NP-Complete P-SPACE NP P July 1, 2008 IUCEE: Discrete Mathematics 15