TSP Cuts Which Do Not Conform to the Template Paradigm

The first computer implementation of the Dantzig-FulkersonJohnson cutting-plane method for solving the traveling salesman problem, written by Martin, used subtour inequalities as well as cutting planes of Gomory’s type. The practice of looking for and using cuts that match prescribed templates in conjunction with Gomory cuts was continued in computer codes of Miliotis, Land, and Fleischmann. Grötschel, Padberg, and Hong advocated a different policy, where the template paradigm is the only source of cuts; furthermore, they argued for drawing the templates exclusively from the set of linear inequalities that induce facets of the TSP polytope. These policies were adopted in the work of Crowder and Padberg, in the work of Grötschel and Holland, and in the work of Padberg and Rinaldi; their computer codes produced the most impressive computational TSP successes of the nineteen eighties. Eventually, the template paradigm became the standard frame of reference for cutting planes in the TSP. The purpose of this paper is to describe a technique for finding cuts that disdains all understanding of the TSP polytope and bashes on regardless of all prescribed templates. Combining this technique with the traditional template approach was a crucial step in our solutions of a 13,509-city TSP instance and a 15,112-city TSP instance. 1 The Cutting-Plane Method and Its Descendants The groundbreaking work of Dantzig, Fulkerson, and Johnson [19] on the traveling salesman problem introduced the cutting-plane method , which can be used to attack any problem minimize cx subject to x ∈ S, (1) where S is a finite subset of some Euclidean space IR, provided that an efficient algorithm to recognize points of S is available. This method is iterative; each Supported by ONR Grant N00014-01-1-0058 M. Jünger and D. Naddef (Eds.): Computat. Comb. Optimization, LNCS 2241, pp. 261–303, 2001. c © Springer-Verlag Berlin Heidelberg 2001 262 D. Applegate et al. of its iterations begins with a linear programming relaxation of (1), meaning a problem minimize cx subject to Ax ≤ b, (2) where the polyhedron P defined as {x : Ax ≤ b} contains S and is bounded. Since P is bounded, we can find an optimal solution x∗ of (2) which is an extreme point of P . If x∗ belongs to S, then it constitutes an optimal solution of (1); otherwise, some linear inequality separates x∗ from S in the sense of being satisfied by all the points in S and violated by x∗; such an inequality is called a cutting plane or simply a cut . In the latter case, we find a family of cuts, add them to the system Ax ≤ b, and use the resulting tighter relaxation of (1) in the next iteration of the cutting-plane method. Each iteration of the method requires first finding x∗ and then finding a family of cuts. Finding x∗ presents no problem: this is what the simplex method and other LP algorithms are for. Finding cuts is the challenge that has to be answered with each new application of the cutting-plane method; we shall return to this challenge later. Progress of the cutting-plane method towards solving a particular instance of problem (1) is often estimated by the increase in the optimal value of its LP relaxation; as more and more cuts are added, these increases tend to get smaller and smaller. When they become unbearably small, the sensible thing to do may be to branch: having chosen a vector α and numbers β′, β′′ with β′ < β′′ such that αTx∗ ∈ (β′, β′′) and {αTx : x ∈ S} ⊂ (−∞, β′] ∪ [β′′,+∞), we solve the two subproblems, minimize cx subject to x ∈ S, αx ≤ β′ and minimize cx subject to x ∈ S, αx ≥ β′′, separately. (If all the elements of S are integer vectors and some component xe of x∗ is not an integer, then we may choose α so that αx is identically equal to xe and set β′ = xe , β′′ = xe .) At some later time, one or both of these two subproblems may be split into sub-subproblems, and so on; in the resulting binary tree of subproblems, each node has the form minimize cx subject to x ∈ S, Cx ≤ d (3) for some system Cx ≤ d of linear inequalities and each leaf will have been either solved without recourse to branching or else found irrelevant since the optimal value of its LP relaxation turned out to be at least as large as cx for some previously known element x of S. This scheme is one of the many variants of the branch-and-bound method. (The term “branch-and-bound”, coined by Little, Murty, Sweeney, and Karel [44], refers to a general class of algorithms that originated in the work of Bock [7], Croes [16], Eastman [20], Rossman and Twery TSP Cuts Which Do Not Conform to the Template Paradigm 263 [65], and Land and Doig [42]; in this more general context, relaxations of (1) may come from a universe far wider than that of linear programming relaxations (2) and each subproblem may be split into more than two sub-subproblems.) Computer codes written by Hong [39], Miliotis [48], and Grötschel, Jünger, and Reinelt [33] introduced a particular variant of this variant, where each subproblem is attacked by the cutting-plane method; in these codes, the cuts introduced in solving (3) are satisfied by all points of S (rather than merely by all points x of S which satisfy Cx ≤ d), and so they can be passed to any other subproblem later on. Padberg and Rinaldi [60] termed this approach branchand-cut . 2 Ways of Finding Cuts The symmetric traveling salesman problem, or TSP for short, is this: given a finite number of “cities” along with the cost of travel between each pair of them, find the cheapest way of visiting all of the cities and returning to your starting point. The travel costs are symmetric in the sense that traveling from city X to city Y costs just as much as traveling from Y to X; the “way of visiting all the cities” is simply the order in which the cities are visited. This problem is a special case of (1) with m = n(n− 1)/2, where n is the number of the cities and S consists of the set of incidence vectors of all the hamiltonian cycles through the set V of the n cities; in this context, hamiltonian cycles are commonly called tours. Dantzig, Fulkerson, and Johnson illustrated the power of their cuttingplane method by solving a 49-city instance of the TSP, an impressive size at the time. They let the initial polyhedron P consist of all vectors x, with components subscripted by edges of the complete graph on V , that satisfy 0 ≤ xe ≤ 1 for all edges e (4) and ∑ (xe : v ∈ e) = 2 for all cities v. (5) (Throughout this paper, we treat the edges of a graph as two-element subsets of its vertex-set: v ∈ e means that vertex v is an endpoint of edge e; e ∩Q = ∅ means that edge e has an endpoint in set Q; e − Q = ∅ means that edge e has an endpoint outside set Q; and so on.) All but two of their cuts have the form ∑ (xe : e ∩ Q = ∅, e − Q = ∅) ≥ 2, where Q is a nonempty proper subset of V ; they are satisfied by all tours through V because every such tour has to move from Q to V − Q at least once and it has to move back to Q after each such crossing. Dantzig, Fulkerson, and Johnson called such inequalities “loop constraints”; nowadays, they are commonly referred to as “subtour elimination inequalities”; we are going to call them simply subtour inequalities. (As for the two exceptional cuts, Dantzig, Fulkerson, and Johnson give ad hoc combinatorial arguments to show that these inequalities are satisfied by incidence vectors of all tours through the 49 cities and, in a footnote, they say “We are indebted to I. Glicksberg of Rand for pointing out relations of this kind to us”.) 264 D. Applegate et al. An important class of problems (1) are the integer linear programming problems, where S is specified as the set of all integer solutions of some explicitly recorded system of linear constraints. For this class, Gomory [26,27,28] designed fast procedures for generating cuts from the optimal simplex basis (and proved that systematic use of these cuts makes the cutting-plane method terminate); cuts generated by these procedures are called Gomory cuts. If an LP relaxation of a TSP instance includes all constraints (4), (5), then a nonempty set of cuts can be found routinely whenever x∗ ∈ S: on the one hand, if x∗ is not an integer vector, then it violates a Gomory cut; on the other hand, if x∗ is an integer vector, then it is the incidence vector of the edge-set of a disconnected graph and each connected component of this graph yields a subtour cut. The first computer code for solving the TSP by the cutting-plane method, written by Martin [46], adopts this policy: some of its cuts are subtour inequalities and others are generated by a variation on Gomory’s theme described in Martin [45]. In subsequent TSP codes, subtour inequalities became a stock item, but Gomory cuts fell into disuse when a different paradigm for finding cuts took over. By a template, we mean a set of linear inequalities; we say that a cut matches the template if it belongs to the set. By the template paradigm, we mean the following two-part procedure used in the design of branch-and-cut algorithms: (i) describe one or more templates of linear inequalities that are satisfied by all the points of S, (ii) for each template described in part (i), design an efficient separation algorithm that, given an x∗, attempts to find a cut that matches the template. The separation algorithms in (ii) may be exact in the sense of finding a cut that separates x∗ from S and matches the template whenever one exists and they may be heuristic in the sense of sometimes failing to find such a cut even though one exists. The primordial template of TSP cuts is the set of subtour inequalities; an exact separation algorithm for this template has been pointed out by Hong [39]. Next came the template of “blossom inequalities”, introduced by Edmonds [21] in the context of 2-matchings and used in the branch-and-cut TSP computer code written by Hong [39]; then came the more general template of “comb inequalities”, first used by Grötschel [30,31] in his solution of a 120-city TSP instance by the cutting-plane method. To describe these templates, let us define, for every vector x with components subscripted by edges of the complete graph on V and for every pair A,B of disjoint subsets of V , x(A,B) = ∑ (xe : e ∩A = ∅, e ∩B = ∅). In this notation, subtour inequalities are recorded as x(Q,V − Q) ≥ 2; a comb inequality is any inequality