24.6. 24.6 Branch and Cut

Branch and Cut (B&C) in the simplest case is a B&B method such that the a certain kind of information is collected and used during the whole course of the algorithm. The theoretical background is based on the notion of integer hull

Definition 24.9 Let

be a polyhedral set where is an matrix, and are and dimensional vectors. All elements of and are rationals. The convex hull of the integer points of is called the integer hull of , i.e. it is the set

The integer hull of the polyhedral set of problem (24.36) is the convex hull of the points (0,0), (0,3), (1,2), and (1,1) as it can be seen on Figure 24.2. Thus the description of the integer hull as a polyhedral set is the inequality system:

Under the conditions of the definition the integer hull is a polyhedral set, too. It is a non-trivial statement and in the case of irrational coefficients it can be not true. If the integer hull is known, i.e. a set of linear inequalities defining exactly the integer hull polyhedral set is known, then the integer programming problem can be reduced to a linear programming problem. Thus problem (24.36) is equivalent to the problem

As the linear programming problem easier to solve than the integer programming problem, one may think that it worth to carry out this reduction. It is not completely true. First of all the number of the linear constraint can be extremely high. Thus generating all constraints of the integer hull can be more difficult than the solution of the original problem. Further on the constraints determining the shape of the integer hull on the side opposite to the optimal solution are not contributing to the finding of the optimal solution. For example the optimal solution of (24.74) will not change if the first constraint is deleted and it is allowed both and may take negative values.

On the other hand the first general integer programming method is the cutting plane method of Gomory. Its main tool is the cut which is based on the observation that possible to determine linear inequalities such that they cut the non-integer optimal solution of the current LP relaxation, but they do not cut any integer feasible solution. A systematic generation of cuts leads to a finite algorithm which finds an optimal solution and proves its optimality if optimal solution exist, otherwise it proves the non-existence of the optimal solution. From geometrical point of view the result of the introducing of the cuts is that the shape of the polyhedral set of the last LP relaxation is very similar to the integer hull in the neighborhood of the optimal solution.

There is the generalization of Gomory's cut called Chvátal (or Chvátal-Gomory) cut. If the two inequalities of (24.36) are summed such that both have weight then the constraint

is obtained. As must be integer the inequality

follows immediately. It is not an algebraic consequence of the original constraints. To obtain it the information of the integrality of the variables had to be used. But the method can be continued. If (24.75) has weight and the second constraint of (24.36) has weight then

is obtained implying

If the last inequality has weight and the first inequality of (24.36) has weight then the result is

implying

Finally the integer hull is obtained. In general the idea is as follows. Assume that a polyhedral set is defined by the linear inequality system

Let be a vector such that is an integer vector and is a noninteger value. Then

is a valid cut, i.e. all integer points of the polyhedral set satisfy it. As a matter of fact it can be proven that a systematic application of the method creates a complete description of the integer hull after finite many steps.

The example shows that Gomory and Chvátal cuts can help to solve a problem. On the other hand they can be incorporated in a B&B frame easily. But in the very general case it is hopeless to generate all effective cuts of this type.

The situation is significantly different in the case of many combinatorial problems. There are many theoretical results known on the type of the facet defining constraints of special polyhedral sets. Here only one example is discussed. It is the Traveling Salesperson Problem (TSP). A salesman must visit some cities and at the end of the tour he must return to his home city. The problem is to find a tour with minimal possible length. TSP has many applications including cases when the “cities” are products or other objects and the “distance” among them doesn't satisfy the properties of the geometric distances, i.e. symmetry and triangle inequality may be violated.

The first exact mathematical formulation of the problem is the so-called Dantzig-Fulkerson-Johnson (DFJ) model. DFJ is still the basis of the numerical solutions. Assume that the number of cities is . Let the distance of the route from city to city . DFJ uses the variables such that

The objective function is the minimization on the total travel length:

The set of the constraints consists of three parts. The meaning of the first part is that the salesman must travel from each city to another city exactly once:

The second part is very similar. It claims that the salesman must arrive to each city from somewhere else again exactly once:

Constraints (24.78) and (24.79) are the constraints of an assignment problem. Taking into account that the variables must be binary Problem (24.77)-(24.79) is really an assignment problem. They don't exclude solutions consisting of several smaller tours. For example if and and then all other variables must be zero. The solution consists of two smaller tours. The first one visits only cities 1, 2, and 3, the second one goes through the cities 4, 5, and 6. The small tours are called subtours in the language of the theory of TSP.

Thus further constraints are needed which excludes the subtours. They are called subtour elimination constraints. There are two kinds of logic how the subtours can be excluded. The first one claims that in any subset of the cities which has at least two elements but not the complete set of the cities the number of travels must be less than the number of elements of the set. The logic can be formalized as follows:

The other logic claims that the salesman must leave all such sets. Let . Then the subtour elimination constraints are the inequalities

The numbers of the two types of constraints are equal and exponential. Although the constraints (24.78)–(24.80) or (24.78), (24.79), and (24.81) are satisfied by only binary vectors being characteristic vectors of complete tours but the polyhedral set of the LP relaxation is strictly larger than the integer hull.

On the other hand it is clear that it is not possible to claim all of the subtour elimination constraints in the real practice. What can be done? It is possible to claim only the violated once. The difficulty is that the optimal solution of the LP relaxation is a fractional vector in most of the cases and that subtour elimination constraint must be found which is violated by the fractional solution provided that such constraint exists as the subtour elimination constraints are necessary to the description of the integer hull but further constraints are needed, too. Thus it is possible that there is no violated subtour elimination constraint but the optimal solution of the LP relaxation is still fractional.

To find a violated subtour elimination constraint is equivalent to the finding of the absolute minimal cut in the graph which has only the edges having positive weights in the optimal solution of the relaxed problem. If the value of the absolute minimal cut is less than 1 in the directed case or less than 2 in the non-directed case then such a violated constraint exists. The reason can be explained based on the second logic of the constraints. If the condition is satisfied then the current solution doesn't leaves at least one of the two sets of the cut in enough number. There are many effective methods to find the absolute minimal cut.

A general frame of the numerical solution of the TSP is the following. In a B&B frame the calculation of the lower bound is repeated until a new violated subtour elimination constraint is obtained, that is the new inequality is added to the relaxed problem and the LP optimization is carried out again. If all subtour elimination constraints are satisfied and the optimal solution of the relaxed problem is still non-integer then branching is made according to a fractional valued variable.

The frame is rather general. The violated constraint cuts the previous optimal solution and reoptimization is needed. Gomory cuts do the same for the general integer programming problem. In the case of other combinatorial problems special cuts may work if the description of the integer hull is known.

Thus the general idea of B&C is that a cut is generated until it can be found and the improvement in the lower bound is great enough. Otherwise branching is made by a non-integer variable. If the cut generation is made only at the root of the enumeration tree then the name of the method is Cut and Branch (C&B). If a cut is generated in a branch then it is locally valid in that branch and in its successors. The cuts generated at the root are valid globally, i.e. in all branches. In some cases, e.e. in binary optimization, it is possible to modify it such that it is valid in the original problem, too.

For practical reasons the type of the generated cut can be restricted. It is the case in TSP as the subtour elimination constraints can be found relatively easily.