A Conjecture Related to the Traveling Salesman Problem

Abstract

We show that certain ways of solving some combinatorial optimization problems can be understood as using query planes to divide the space of problem instances into polyhedra that could fit into those that characterize the problem's various solutions. This viewpoint naturally leads to a splinter-proneness property that is then shown to be responsible for the hardness of the concerned problem. We conjecture that the $NP$-equivalent traveling salesman problem (TSP) has this property and hence is hard to solve to a certain extent.