Years ago, when I was in high school and college, the board games Risk and Diplomacy served as wonderful playing fields where I could develop and exercise my tactical and negotiating skills.
One issue that often came up in my forays into international intrigue was how to deploy my limited forces to defend far-flung territories while I plotted to conquer the world.
Such questions of military strategy can be handled mathematically. Charles S. ReVelle of Johns Hopkins University in Baltimore and Kenneth E. Rosing, recently retired from Erasmus University Rotterdam, describe their approach in the August-September American Mathematical Monthly.
ReVelle is a pioneer in the relatively new and rapidly growing field of location science, which involves the use of mathematical techniques to determine the best places to put fire stations, hospitals, fast-food restaurants, or even military units.
One classic case goes back to the years of the Roman empire’s decline. In the 3rd century, Rome dominated Europe and could station 50 legions throughout its territories, securing even the most distant lands. A century later, it had only about half as many legions to defend the empire.
Emperor Constantine (274–337) had to decide how to disperse the legions at his disposal to protect the empire’s fringes without leaving Rome itself open to attack. He devised a “defense-in-depth” strategy to cope with the diminished power of his forces. He expected local, part-time militias to slow down and fragment invading barbarian hordes, and he dispatched well-equipped and highly trained mobile field armies, when needed, to halt or throw back the enemy or suppress an insurrection.
Constantine organized his legions into four field armies. He had eight regions where he could station these forces (see map). The trick was to place the armies so that each region was secured–at least one army was present–or securable–an army was just one “step” away. However, an army could be sent from one region to an adjacent region only if it moved from a region where there was at least one other army present to help launch its move.
Constantine chose to place two armies in Rome and two at his new capital, Constantinople. With this strategy, only Britain could not be secured or reached in one step. Defending Britain would require the movement of an army from Rome to Gaul, securing Gaul, followed by the movement of a second army from Constantinople to Rome, then from Rome to Gaul, and finally to Britain–a total of four steps. “It is no wonder that Britain was lost,” ReVelle remarks (see http://www.jhu.edu/~jhumag/0497web/locate3.html).
Other placements are possible. You could, for example, put one army in Gaul, two in Rome, and one in Constantinople. This deployment, however, has its own flaws.
ReVelle decided to see if he could improve on Constantine’s solution–either by reducing the number of regions that can’t be reached in one step (from one to zero, in this case) or by cutting the number of steps it takes to reach the worst-off region.
Two fundamental mathematical questions underlie Constantine’s problem. What is the smallest number of field armies needed to defend all the regions? If the number available is less than this minimum, how should the limited resources be deployed to defend the largest number of provinces?
These formulations put the military problem in the realm of location science and combinatorial optimization techniques.
Using linear programming, ReVelle came up with several alternative deployments for Constantine’s forces. His “Roman solution” places two armies in Rome, one in Britain, and one in Asia Minor. However, “this deployment suffers from a reduced capability to respond to a second war occurring somewhere else in the empire,” ReVelle and Rosing note.
Keeping in mind the possibility of the need to open a second front, you could put two armies in Iberia and two in Egypt. This solution, however, raises other issues, including the political one of leaving no army in Rome itself.
In general, what constitutes an acceptable answer depends very strongly on the constraints that you impose in defining the problem.
Nonetheless, “the reduction of [a famous military strategy] puzzle to a mathematical programming form suggests the possibility that other puzzles–some of which may have significant applications–have related types of formulations,” ReVelle and Rosing conclude.