W. I. B. Beveridge, in his classic guidebook for young scientists [7], likens scientific research “to warfare against the unknown”:

This chapter is about developing small- and large-scale plans of attack in algorithmic experiments.

To make the discussion concrete, we consider algorithms for the graph coloring (GC) problem. The input is a graph G containing n vertices and m edges. A coloring of G is an assignment of colors to vertices such that no two adjacent vertices have the same color. Figure 2.1 shows an example graph with eight vertices and 10 edges, colored with four colors. The problem is to find a coloring that uses a minimum number of colors – is 4 the minimum in this case?

When restricted to planar graphs, this is the famous map coloring problem, which is to color the regions of a map so that adjacent regions have different colors. Only four colors are needed for any map, but in the general graph problem, as many as n colors may be required.