A fundamental problem in linear inequalities with applications to the travelling salesman problem

We consider a system of m linearly independent equality constraints in n nonnegat ive variables: A x = b, x >= O. The fundamenta l problem that we discuss is the following: suppose we are given a set o f r linearly independent co lumn vectors o f A, known as the special column vectors. The problem is to develop an efficient algorithm to determine whether there exists a feasible basis which contains all the special co lumn vectors as basic co lumn vectors and to find such a basis if one exists. Such an algorithm has several applications in the area o f mathemat ica l programming. As an illustration, we show that the famous travelling salesman problem can be solved efficiently using this algorithm. Recent published work indicates tha t this algorithm has applications in integer linear programming. An algorithm for this problem using a set covering approach is described.