Map segmentation

In mathematics, the map segmentation problem is a kind of optimization problem. It involves a certain geographic region that has to be partitioned into smaller sub-regions in order to achieve a certain goal. Typical optimization objectives include:
 * Minimizing the workload of a fleet of vehicles assigned to the sub-regions;
 * Balancing the consumption of a resource, as in fair cake-cutting.
 * Determining the optimal locations of supply depots;
 * Maximizing the surveillance coverage.

Fair division of land has been an important issue since ancient times, e.g. in ancient Greece.

Notation
There is a geographic region denoted by C ("cake").

A partition of C, denoted by X, is a list of disjoint subregions whose union is C:
 * $$C = X_1\sqcup\cdots\sqcup X_n$$

There is a certain set of additional parameters (such as: obstacles, fixed points or probability density functions), denoted by P.

There is a real-valued function denoted by G ("goal") on the set of all partitions.

The map segmentation problem is to find:
 * $$\arg\min_X G(X_1,\dots,X_n \mid P)$$

where the minimization is on the set of all partitions of C.

Often, there are geometric shape constraints on the partitions, e.g., it may be required that each part be a convex set or a connected set or at least a measurable set.

Examples
1. Red-blue partitioning: there is a set $$P_b$$ of blue points and a set $$P_r$$ of red points. Divide the plane into $$n$$ regions such that each region contains approximately a fraction $$1/n$$ of the blue points and $$1/n$$ of the red points. Here:
 * The cake C is the entire plane $$\mathbb{R}^2$$;
 * The parameters P are the two sets of points;
 * The goal function G is
 * $$G(X_1,\dots,X_n) := \max_{i\in \{1,\dots, n\}} \left( \left |\frac{|P_b\cap X_i| - |P_b|} n \right| + \left| \frac{|P_r\cap X_i| - |P_r|} n\right| \right).$$
 * It equals 0 if each region has exactly a fraction $$1/n$$ of the points of each color.

Related problems

 * A Voronoi diagram is a specific type of map-segmentation problem.
 * Fair cake-cutting, when the cake is two-dimensional, is another specific map-segmentation problem when the cake is two-dimensional, like in the Hill–Beck land division problem.
 * The Stone–Tukey theorem is related to a specific map-segmentation problem.