For many problems, it is possible to solve small instances but not large instances. Furthermore, it is easy to identify parts of the problem that are largely independent from other parts. The break-combine philosophy suggests breaking the problem into small pieces and solving each one independently. The pieces are then reformed (if necessary) into one large solution.
Again, consider the TSP. If the points are on the plane, with distances Euclidean, it is clear that the upper left points will have little to do with the lower right points. This suggests some sort of break-combine algorithm. Here is one which works well (very well theoretically).
Divide the points in two equal pieces, say vertically. Then divide each piece into two equal pieces horizontally. Continue alternating horizontal and vertical cuts until each piece is small enough to be solve exactly. Solve each piece exactly. This is the break phase.
Now combine pairs of solutions separated by the a line in the reverse order they were divided. Combine them by finding the best pair of edges to add and the best to delete to form a tour of the combined pieces. Continue until all the tours are combined. This is the combine phase.
This method works best when there is a natural division of the problem (for instance, when the problem is embedded in the plane) and when the combine phase does not do too poorly.