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Great Ideas in Computing Complexity Theory

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Title: Great Ideas in Computing Complexity Theory


1
Great Ideas in ComputingComplexity Theory
  • Richard Anderson
  • University of Washington

2
Mathematical Structure of Computation
  • What is computation?
  • Churchs Thesis all forms of computation are
    equivalent

3
Alan Turing
  • Undecidability of the halting problem
  • A function that provably cannot be computed

4
Combinatorial Optimization
  • Ford-Fulkerson 1956

5
Jack Edmonds
  • Polynomial Time

6
Non deterministic Turing Machine
7
Hard problems
  • Satisfiability
  • Bin Packing
  • Integer Programming
  • Hamiltonian Circuit
  • Vertex Cover
  • 3 Dimensional Matching
  • Traveling Salesman Problem

8
NP Completeness
  • Non-deterministic polynomial time
  • Cooks 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

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)
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

11
Gadgets Truth Setting
12
Gadgets Truth Testing
13
Papadimitriou
  • Hamiltonian Circuit NP Complete for Grid Graphs

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

15
On beyond NP
NP-Complete
P-SPACE
NP
P
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