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

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Mathematical Structure of Computation What is computation? Great Ideas in Computing Complexity Theory Richard Anderson Churchs Thesis all forms of computation are equivalent University of Washington July 1, 2008 IUCEE:

SLIDE 1

Great Ideas in Computing Complexity Theory

Richard Anderson University of Washington

July 1, 2008 1 IUCEE: Discrete Mathematics

Mathematical Structure of Computation

What is computation?
Church’s Thesis – all forms of computation

are equivalent

July 1, 2008 IUCEE: Discrete Mathematics 2

Alan Turing

Undecidability of the halting problem
A function that provably cannot be

computed

July 1, 2008 IUCEE: Discrete Mathematics 3

Combinatorial Optimization

Ford-Fulkerson 1956

July 1, 2008 IUCEE: Discrete Mathematics 4

Jack Edmonds

Polynomial Time

July 1, 2008 IUCEE: Discrete Mathematics 5

Non deterministic Turing Machine

July 1, 2008 IUCEE: Discrete Mathematics 6

SLIDE 2

Hard problems

Satisfiability
Bin Packing
Integer Programming
Hamiltonian Circuit
Vertex Cover
3 Dimensional Matching
Traveling Salesman Problem

July 1, 2008 IUCEE: Discrete Mathematics 7

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 8

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

July 1, 2008 IUCEE: Discrete Mathematics 9

(x || y || z) && (!x || !y || !z) && (!x || y) && (x || !y) && (y || !z) && (!y || z)

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 10

Gadgets: Truth Setting

July 1, 2008 IUCEE: Discrete Mathematics 11

Gadgets: Truth Testing

July 1, 2008 IUCEE: Discrete Mathematics 12

SLIDE 3

Papadimitriou

Hamiltonian Circuit NP Complete for Grid

Graphs

July 1, 2008 IUCEE: Discrete Mathematics 13

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 14

On beyond NP

July 1, 2008 IUCEE: Discrete Mathematics 15

P-SPACE NP-Complete NP P