1. Great Ideas in Computing Complexity Theory
Richard Anderson
University of Washington
July 1, 2008
IUCEE: Discrete Mathematics

2. Mathematical Structure of Computation
What is computation?
Church’s Thesis – all forms of computation are equivalent
July 1, 2008
IUCEE: Discrete Mathematics

3. IUCEE: Discrete Mathematics
Alan Turing
Undecidability of the halting problem
A function that provably cannot be computed
July 1, 2008
IUCEE: Discrete Mathematics

4. Combinatorial Optimization
Ford-Fulkerson 1956
July 1, 2008
IUCEE: Discrete Mathematics

5. IUCEE: Discrete Mathematics
Jack Edmonds
Polynomial Time
July 1, 2008
IUCEE: Discrete Mathematics

6. Non deterministic Turing Machine
July 1, 2008
IUCEE: Discrete Mathematics

7. IUCEE: Discrete Mathematics
Hard problems
Satisfiability
Bin Packing
Integer Programming
Hamiltonian Circuit
Vertex Cover
3 Dimensional Matching
Traveling Salesman Problem
July 1, 2008
IUCEE: Discrete Mathematics

8. IUCEE: Discrete Mathematics
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
July 1, 2008
IUCEE: Discrete Mathematics

9. IUCEE: Discrete Mathematics
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)
July 1, 2008
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
July 1, 2008
IUCEE: Discrete Mathematics

11. Gadgets: Truth Setting
July 1, 2008
IUCEE: Discrete Mathematics

12. Gadgets: Truth Testing
July 1, 2008
IUCEE: Discrete Mathematics

13. IUCEE: Discrete Mathematics
Papadimitriou
Hamiltonian Circuit NP Complete for Grid Graphs
July 1, 2008
IUCEE: Discrete Mathematics

14. IUCEE: Discrete Mathematics
Euclidean TSP
n points in a Rn
Distance between a pair of points is the Euclidean distance
Is the Euclidean TSP NP-complete?
July 1, 2008
IUCEE: Discrete Mathematics

15. IUCEE: Discrete Mathematics
On beyond NP
NP-Complete
P-SPACE NP P
July 1, 2008
IUCEE: Discrete Mathematics