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

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.

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.

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.

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

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

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

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

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

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

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.

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.

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

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

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.

Ideal soap solution

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