Integer Programming Formulation of Traveling Salesman Problems

TLDR: The present paper provides yet another example of the versatility of integer programming as a mathematical modeling device by representing a generalization of the well-known “Travelling Salesman Problem” in integer programming terms.

Abstract: It has been observed by many people that a striking number of quite diverse mathematical problems can be formulated as problems in integer programming, that is, linear programming problems in which some or all of the variables are required to assume integral values. This fact is rendered quite interesting by recent research on such problems, notably by R. E. Gomory [2, 3], which gives promise of yielding efficient computational techniques for their solution. The present paper provides yet another example of the versatility of integer programming as a mathematical modeling device by representing a generalization of the well-known “Travelling Salesman Problem” in integer programming terms. The authors have developed several such models, of which the one presented here is the most efficient in terms of generality, number of variables, and number of constraints. This model is due to the second author [4] and was presented briefly at the Symposium on Combinatorial Problems held at Princeton University, April 1960, sponsored by SIAM and IBM. The problem treated is: (1) A salesman is required to visit each of n cities, indexed by 1, … , n. He leaves from a “base city” indexed by 0, visits each of the n other cities exactly once, and returns to city 0. During his travels he must return to 0 exactly t times, including his final return (here t may be allowed to vary), and he must visit no more than p cities in one tour. (By a tour we mean a succession of visits to cities without stopping at city 0.) It is required to find such an itinerary which minimizes the total distance traveled by the salesman.Note that if t is fixed, then for the problem to have a solution we must have tp g n. For t = 1, p g n, we have the standard traveling salesman problem.Let dij (i ≠ j = 0, 1, … , n) be the distance covered in traveling from city i to city j. The following integer programming problem will be shown to be equivalent to (1): (2) Minimize the linear form ∑0li≠jln∑ dijxij over the set determined by the relations ∑ni=0i≠jxij = 1 (j = 1, … , n) ∑nj=0j≠ixij = 1 (i = 1, … , n) ui - uj + pxij l p - 1 (1 l i ≠ j l n) where the xij are non-negative integers and the ui (i = 1, …, n) are arbitrary real numbers. (We shall see that it is permissible to restrict the ui to be non-negative integers as well.)If t is fixed it is necessary to add the additional relation: ∑nu=1xi0 = t Note that the constraints require that xij = 0 or 1, so that a natural correspondence between these two problems exists if the xij are interpreted as follows: The salesman proceeds from city i to city j if and only if xij = 1. Under this correspondence the form to be minimized in (2) is the total distance to be traveled by the salesman in (1), so the burden of proof is to show that the two feasible sets correspond; i.e., a feasible solution to (2) has xij which do define a legitimate itinerary in (1), and, conversely a legitimate itinerary in (1) defines xij, which, together with appropriate ui, satisfy the constraints of (2).Consider a feasible solution to (2).The number of returns to city 0 is given by ∑ni=1xi0. The constraints of the form ∑ xij = 1, all xij non-negative integers, represent the conditions that each city (other than zero) is visited exactly once. The ui play a role similar to node potentials in a network and the inequalities involving them serve to eliminate tours that do not begin and end at city 0 and tours that visit more than p cities. Consider any xr0r1 = 1 (r1 ≠ 0). There exists a unique r2 such that xr1r2 = 1. Unless r2 = 0, there is a unique r3 with xr2r3 = 1. We proceed in this fashion until some rj = 0. This must happen since the alternative is that at some point we reach an rk = rj, j + 1