Mathematics

Biology

Algorithm

Mathematical optimization

Reuse

Travelling salesman problem

Ecology

Computer science

The Traveling Salesman Problem (TSP) was first formulated in 1930 and is one of the most studied problems in optimization. If the optimal solution to the TSP can be found in polynomial time, it would then follow that every NP-hard problem could be solved in polynomial time, proving P=NP. It will be shown that our algorithm finds P~NP with scale. Using a δ –ε proof, it is straightforward to show that as the number of cities goes to infinity, P goes to NP (i.e., ε >0 ). This was demonstrated using a quadratic number of parallel processors because that speedup, by definition, is polynomial. A fastest parallel algorithm is defined. Six distinct 3-D charts of empirical results are supplied. For example, using an arbitrary run of 5,000 cities, we obtained a tour within 0.00001063 percent of the optimal using 4,166,667 virtual processors (Intel Xenon E5-1603 @ 2.8 GHz). To save the calculated extra 209 miles would take a quantum computer, the fastest possible computer, over (5,000!/(2**4,978 * 22!)) * 267,782 centuries. Clearly, the automated acquisition of heuristics and the associated P~NP solutions are an important problem warranting attention.

Reusing the NP-Hard Traveling-Salesman Problem to Demonstrate That P~NP (Invited Paper)