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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 • Quantum computing • 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. Find the minimum spanning tree for a bunch of vertices, given that you can add additional points. Simply put: 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.