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Approaching P=NP: Can Soap Bubbles Solve The Steiner Tree Problem In Polynomial Time?

Long Ouyang Computer systems. Approaching P=NP: Can Soap Bubbles Solve The Steiner Tree Problem In Polynomial Time? . Introduction. Decision problems – Ask yes/no questions. Two classes of problems, P and NP

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Approaching P=NP: Can Soap Bubbles Solve The Steiner Tree Problem In Polynomial Time?

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

  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 • Quantum computing • 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.

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