Divide and conquer orthogonality principle for parallel optimizations in TSP

Abstract

A lossless divide-and-conquer (D & C) principle, implemented via an orthogonality projection, is described and illustrated with a recursive solution of the Traveling Salesman Problem (TSP), which is executable on a massively parallel and distributed computer. The lossless D & C principle guides us to look near the desirable boundary and to seek for a characteristic vector V (e.g. displacement vector in TSP) such that V ≈ A + B where two resultant vectors A and B (located one in each sub-domains to be divided) should be orthogonal to each other having no cross product terms, i.e. (A, B) = (B, A) = 0. In the case of parallel computing this goal amounts to minimum communication cost among processors. Then the global optimization of the whole domain: Min. ∥ V∥ 2 = Min. ∥ A∥ 2 + Min. ∥ B∥ 2 can be losslessly divided into two sub-domains that each can seek its own optimized solution separately (by two separate sets of processors without inter-communication during processing). Such a theorem of orthogonal division error (ODE) for lossless D & C is proved, and the orthogonal projection is constructed for solving a large-scale TSP explicitly.