Traveling Salesman Polytope Research Articles

Small Travelling Salesman Polytopes

We introduce a new class of valid inequalities for the polytope of the symmetric travelling salesman problem. We also gave complete characterizations of the polytope for 6 and 7 cities. For the latter case, the new inequalities are needed. These results are related to work of R. Z. Norman in the 1950's.

Facets of the Asymmetric Traveling Salesman Polytope

In this paper we consider the Asymmetric Traveling Salesman polytope, Pn, defined as the convex hull of the incidence vectors of tours in a complete digraph with n vertices, and its monotonization, P̃n. Several classes of valid inequalities for both there polytopes have been introduced in the literature which are not known to define facets of Pn or P̃n. We describe a general technique which can often be used to prove that a given inequality defines a facet of Pn. The method is applied to prove that the so-called Dk+, Dk−, C3, comb and C2 inequalities define facets of Pn, thus solving open problems. As for the monotone polytope, a necessary and sufficient condition for a facet-defining inequality of Pn to define a facet of P̃n as well, is introduced which applies to the Dk+, Dk−, C3, comb and C2 inequalities. In addition, a simple procedure for transforming any facet-defining inequality of Pn into an equivalent one which also defines a facet of P̃n, is given.

Facet identification for the symmetric traveling salesman polytope

Several procedures for the identification of facet inducing inequalities for the symmetric traveling salesman polytope are given. An identification procedure accepts as input the support graph of a point which does not belong to the polytope, and returns as output some of the facet inducing inequalities violated by the point. A procedure which always accomplishes this task is calledexact, otherwise it is calledheuristic. We give exact procedures for the subtour elimination and the 2-matching constraints, based on the Gomory—Hu and Padberg—Rao algorithms respectively. Efficient reduction procedures for the input graph are proposed which accelerate these two algorithms substantially. Exact and heuristic shrinking conditions for the input graph are also given that yield efficient procedures for the identification of simple and general comb inequalities and of some elementary clique tree inequalities. These procedures constitute the core of a polytopal cutting plane algorithm that we have devised and programmed to solve a substantial number of large-scale problem instances with sizes up to 2392 nodes to optimality.

The Asymmetric Assignment Problem and Some New Facets of the Traveling Salesman Polytope on a Directed Graph

An assignment (spanning union of node-disjoint dicycles) in a directed graph is called asymmetric if it contains at most one arc of each pair $(i, j)$, $( j,i)$. The asymmetric assignment polytope is the convex hull of the incidence vectors of all asymmetric assignments. A class of facets is described for this polytope defined on a complete digraph, associated with certain odd-length closed alternating trails. The inequalities defining these facets are also facet inducing for the traveling salesman polytope defined on the same digraph. Furthermore, this class of facets is distinct from each of the classes identified earlier.

The facets of the asymmetric 5-city traveling salesman polytope

The asymmetric 5-TS-Polytope has 390 facets. The different constraint classes are indicated.

Clique Tree Inequalities and the Symmetric Travelling Salesman Problem

The linear programming cutting plane approach for solving the travelling salesman problem has recently proven to be highly successful, cf. Crowder and Padberg (Crowder, H. P., M. W. Padberg. 1980. Solving large-scale symmetric travelling salesman problems to optimality. Management Sci. 26 495–509.), Grötschel (Grötschel, M. 1980a. On the symmetric travelling salesman problem: Solution of a 120 city problem. Math. Programming Stud. 12 61–77.), Padberg and Hong (Padberg, M. W., S. Hong. 1980. On the symmetric travelling salesman problem: A computational study. Math. Programming Stud. 12 78–107.). One of the reasons for this success is certainly the fact that instead of ordinary cutting planes (Gomory-cuts etc.) problem-specific cutting planes could be used which define facets of the underlying integer programming polytopes. In this paper we shall define a new class of inequalities (clique tree inequalities) valid for the travelling salesman polytope which properly contains many of the known classes of inequalities (like subtour elimination constraints, 2-matching constraints, comb inequalities), and we show that all these new inequalities induce facets of the travelling salesman polytope. Since the general structure of these new inequalities is quite simple we hope that it will be possible to use the inequalities efficiently in cutting plane procedures for the travelling salesman problem.

The traveling salesman problem in graphs with 3-edge cutsets

This paper analyzes decomposition properties of a graph that, when they occur, permit a polynomial solution of the traveling salesman problem and a description of the traveling salesman polytope by a system of linear equalities and inequalities. The central notion is that of a 3-edge cutset, namely, a set of 3 edges that, when removed, disconnects the graph. Conversely, our approach can be used to construct classes of graphs for which there exists a polynomial algorithm for the traveling salesman problem. The approach is illustrated on two examples, Halin graphs and prismatic graphs.

Halin graphs and the travelling salesman problem

A Halin graphH=T∪C is obtained by embedding a treeT having no nodes of degree 2 in the plane, and then adding a cycleC to join the leaves ofT in such a way that the resulting graph is planar. These graphs are edge minimal 3-connected, hamiltonian, and in general have large numbers of hamilton cycles. We show that for arbitrary real edge costs the travelling salesman problem can be polynomially solved for such a graph, and we give an explicit linear description of the travelling salesman polytope (the convex hull of the incidence vectors of the hamilton cycles) for such a graph.

On the structure of the monotone asymmetric travelling salesman polytope I: hypohamiltonian facets

The monotone asymmetric travelling salesman polytope P̄ n T is defined to be the convex hull of the incidence vectors of all hamiltonian circuits and all subsets of these in a complete diagraph of order n. We prove that certain hypohamiltonian diagraphs G=( V, E), i.e. diagraphs which are not hamiltonian but such that G–υ is hamiltonian for all υϵ V, induce facets x( E)⩽ n–1 of P̄ n T. This result indicates that P̄ n T has very complicated facets and that it is very unlikely that an explicit complete characterization of P̄ n T can ever be given.

On the Monotone Symmetric Travelling Salesman Problem: Hypohamiltonian/Hypotraceable Graphs and Facets

A graph G is called hypohamiltonian (hypotraceable) if it does not contain a hamiltonian cycle (chain) but if every vertex-deleted subgraph G − v contains a hamiltonian cycle (chain). It is shown that certain classes of these graphs induce facets of the monotone symmetric travelling salesman polytope, i.e. the convex hull of the incidence vectors of all tours and subsets of tours. These results indicate that it is quite unlikely that an explicit complete characterization of this polytope can be obtained.

On the symmetric travelling salesman problem II: Lifting theorems and facets

Four lifting theorems are derived for the symmetric travelling salesman polytope. They provide constructions and state conditions under which a linear inequality which defines a facet of then-city travelling salesman polytope retains its facetial property for the (n + m)-city travelling salesman polytope, wherem ≥ 1 is an arbitrary integer. In particular, they permit a proof that all subtour-elimination as well as comb inequalities define facets of the convex hull of tours of then-city travelling salesman problem, wheren is an arbitrary integer.

On the symmetric travelling salesman problem I: Inequalities

We investigate several classes of inequalities for the symmetric travelling salesman problem with respect to their facet-defining properties for the associated polytope. A new class of inequalities called comb inequalities is derived and their number shown to grow much faster with the number of cities than the exponentially growing number of subtour-elimination constraints. The dimension of the travelling salesman polytope is calculated and several inequalities are shown to define facets of the polytope. In part II (“On the travelling salesman problem II: Lifting theorems and facets”) we prove that all subtour-elimination and all comb inequalities define facets of the symmetric travelling salesman polytope.

The adjacency relation on the traveling salesman polytope is NP-Complete

We consider the problem of determining whether two traveling salesman tours correspond to non-adjacent vertices of the convex polytope associated with the traveling salesman problem. This problem is shown to be NP-Complete for both the symmetric and nonsymmetric traveling salesman problem. Several implications are discussed.

Partial linear characterizations of the asymmetric travelling salesman polytope

We consider the linear programming formulation of the asymmetric travelling salesman problem. Several new inequalities are stated which yield a sharper characterization in terms of linear inequalities of the travelling salesman polytope, i.e., the convex hull of tours.