1. Long Ouyang Computer systems
Approaching P=NP: Can Soap Bubbles Solve The Steiner Tree Problem In Polynomial Time?
Long Ouyang
Computer systems

2. Introduction Decision problems – Ask yes/no questions.
Two classes of problems, P and NP
P: Problems that can be solved in time polynomial to the size of the input by a deterministic Turing machine.
NP: Problems that can be solved in time polynomial to the size of the input by a nondeterministic Turing machine.

3. Turing machines (not important)
Deterministic:
-At most one entry for each combination of symbol and state.
Non-deterministic:
-More than one entry for each combination of symbol and state.

4. What does this mean? With regards to modern computers:
Problems in P can be solved in polynomial time.
Solutions to problems in NP can be verified in polynomial time.
Problems in P take relatively less time to solve, problems in NP take relatively more.

5. NP Problems in NP: Traveling salesman problem Hamiltonian path problem
Partition problem
Multiprocessor scheduling
Bin packing
Sudoku
Tetris

6. Who cares? If P=NP, hard problems are actually relatively easy.
Implications: Cryptography, Mapquest, compression, scheduling, computation

7. How? Try to devise P algorithms to NP-Complete problems.
Problem: Turing arguments, Razborov-Rudich barrier

8. So what do we do?
Physical systems – often in nature, physical systems reduce a situation to its lowest energy state (optimizing energy).
Soap films
Spin glasses
Folding proteins
Bubbles

9. Additional methods Quantum computing
Using DNA as non-deterministic Turing machines.
Time travel
Anthropic principles

10. We’ll take the soap, please
Pros:
It’s inexpensive, compared to time travel.
Reduces P=NP to a problem in digital physics.
Cons:
Makes formal proof at the least, very difficult
Optimistically, at best, provides experimental run-time data

11. The Steiner Problem
Soap is rumored to solve the Steiner Tree Problem (STP).
Steiner Tree Problem:
Description: Given a weighted graph G, G(V,E,w), where V is the set of vertices, E is the set of edges, and w is the set of weights, and S, a subset of V, find the subset of G that contains S and has the minimum weight.
Simply put: Find the minimum spanning tree for a bunch of vertices, given that you can add additional points.

12. How does soap do this?
Soap, in water, acts as a surfactant, which decreases the surface tension of the water.
This acts to minimize the surface energy of the liquid.
This should minimize surface area (graph weight), and solve the problem.

13. Tools used OpenFOAM (computational fluid physics engine)
Paraview (visualization engine)
GeoSteiner '96 (exact STP solver)

14. Design Generation of random vertices, appropriate mesh for OpenFOAM
Solution of STP (where nodes are the random vertices) by GeoSteiner '96
OpenFOAM computation of soap action on vertices
Comparison of exact solution with soap solution

15. Soap model Thin box filled with soap water.
Pegs connect the same parallel faces of the box (nodes)
There's a small drain at the bottom of the box.

16. Ideal soap solution

17. Conclusions Agent-based modeling sucks for modeling fluids.
Rigid-body physics sucks for modeling fluids.