Before examining CBC in more detail, we tersely describe the basic branch-and-cut algorithm by way of example, (which should really be called branch-and-cut-and-bound) and show the major C++ class(es) in CBC related to each step. The major CBC classes, labeled (A) through (F), are described in Table 1.1.
Step 1. (Bound) Given a MIP model to minimize where some variables must take on integer values (e.g., 0, 1, or 2), relax the integrality requirements (e.g., consider each "integer" variable to be continuous with a lower bound of 0.0 and an upper bound of 2.0). Solve the resulting linear model with an LP solver to obtain a lower bound on the MIP's objective function value. If the optimal LP solution has integer values for the MIP's integer variables, we are finished. Any MIP-feasible solution provides an upper bound on the objective value. The upper bound equals the lower bound; the solution is optimal.
Step 2. (Branch) Otherwise, there exists an "integer" variable with a non-integral value. Choose one non-integral variable (e.g., with value 1.3) (A)(B) and branch. Create two ^{[2]} nodes, one with the branching variable having an upper bound of 1.0, and the other with the branching variable having a lower bound of 2.0. Add the two nodes to the search tree.
While (search tree is not empty) {
Step 3. (Choose Node) Pick a node off the tree (C)(D)
Step 4. (Re-optimize LP) Create an LP relaxation and solve.
Step 5. (Bound) Interrogate the optimal LP solution, and try to prune the node by one of the following.
Step 6. (Branch) If we were unable to prune the node, then branch. Choose one non-integral variable to branch on (A)(B). Create two nodes and add them to the search tree. }
This is the outline of a "branch-and-bound" algorithm. If in optimizing the linear programs, we use cuts to tighten the LP relaxations (E)(F), then we have a "branch-and-cut" algorithm. (Note, if cuts are only used in Step 1, the method is called a "cut-and-branch" algorithm.)
Table 1.1. Associated Classes
Note | Class name | Description |
---|---|---|
(A) | CbcBranch... | These classes define the nature of MIP's discontinuity. The simplest discontinuity is a variable which must take an integral value. Other types of discontinuities exist, e.g., lot-sizing variables. |
(B) | CbcNode | This class decides which variable/entity to branch on next. Even advanced users will probably only interact with this class by setting CbcModel parameters ( e.g., priorities). |
(C) | CbcTree | All unsolved models can be thought of as being nodes on a tree where each node (model) can branch two or more times. The interface with this class is helpful to know, but the user can pretty safely ignore the inner workings of this class. |
(D) | CbcCompare... | These classes are used in determine which of the unexplored nodes in the tree to consider next. These classes are very small simple classes that can be tailored to suit the problem. |
(E) | CglCutGenerators | Any cut generator from CGL can be used in CBC. The cut generators are passed to CBC with parameters which modify when each generator will be tried. All cut generators should be tried to determine which are effective. Few users will write their own cut generators. |
(F) | CbcHeuristics | Heuristics are very important for obtaining valid solutions quickly. Some heuristics are available, but this is an area where it is useful and interesting to write specialized ones. |
There are a number of resources available to help new CBC users get started. This document is designed to be used in conjunction with the files in the Samples subdirectory of the main CBC directory (COIN/Cbc/Samples). The Samples illustrate how to use CBC and may also serve as useful starting points for user projects. In the event that either this document or the available Doxygen content conflicts with the observed behavior of the source code, the comments in the header files, found in COIN/Cbc/include, are the ultimate reference.
^{[2] } The current implementation of CBC allow two branches to be created. More general number of branches could be implemented.