Approaching PNP: Can Soap Bubbles Solve The Steiner Tree Problem In Polynomial Time

1
Approaching PNP 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 PNP, 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
• Quantum computing
• Anthropic principles

10
Well take the soap, please

• Pros
• Its inexpensive, compared to time travel.
• Reduces PNP 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.