How important are branching decisions: Fooling MIP solvers

We show the importance of selecting good branching variables by exhibiting a family of instances for which an optimal solution is both trivial to find and provably optimal by a fixed-size branch-and-bound tree, but for which state-of-the-art Mixed Integer Programming solvers need an increasing amount of resources. The instances encode the edge-coloring problem on a family of graphs containing a small subgraph requiring four colors, while the rest of the graph requires only three.