## Branch-and-Cut Overview

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  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.

• LP is infeasible, prune the node.
• Else, the optimal LP solution value of the node exceeds the current upper bound, prune the node.
• Else, the optimal LP solution of the node does not exceed the current upper bound and the solution is feasible to the MIP. Update the upper bound, and the best known MIP solution, and prune the node by optimality.

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.)

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.

 The current implementation of CBC allow two branches to be created. More general number of branches could be implemented.