Multi-Terminal Cuts - Applications and Verifying validity of an approximation algorithm
The multi-terminal cut problem (which is also known as the multiway cut problem) has many important applications. In the more general setting, each edge $e$ has a cost $c_e \ge 0$. The goal is to remove a minimum cost set of edges such that in the resulting graph there is no path between any two vertices in $T$. Here is an application in distributed computing.

Each vertex represents an object, and an edge $e$ of cost $c_e$ between them represents the cost of communication between the objects. The objects have to be partitioned to reside on $k$ different machines, with special object $s_i$ residing on the $i^{th}$ machine. The goal is to partition the objects residing on the $k$ machines in such a way that the communication cost between the machines is minimized.