The algorithm for the traveling salesman problem (TSP) with distances a and b is a polynomial time algorithm. The time required by the algorithm grows with n, the size of the input, as n raised to the power of 2. The algorithm, not only, finds an optimal route, it can find all optimal routes, in a systematic way.
In the general case, every node is connected by an edge to every other node and the cost of every edge is either a or b. The Hamilton cycle problem, a more restricted case of the TSP, asks whether a tour of the graph exists. We can obtain an instance of the TSP with distances a and b from an instance of the Hamilton cycle problem in a straightforward way.
Transforming the factorization of RSA260 to the (undirected) Hamilton cycle problem yields an instance with 55,614,837 nodes and 84,899,139 edges. (an optimal tour will encode the two factors). Such a graph is a sparse graph. Solving the corresponding instance of the TSP with distances a and b, in contrast, is limited by the required amount of memory.
The algorithm can solve the Hamilton cycle problem directly. The corresponding weighted graph is provided as input. In this case, the amount of space and the amount of time needed by the algorithm can be drastically reduced. Thus, in practice, large instances of factoring and of other NP problems can be efficiently solved.
I will publish the factorization of RSA260 within a couple of weeks.
P vs NP 