Question

A cohort of k spies resident in a certain country needs escape routes in case of emergency. They will be travelling using the railway system which we can think of as a directed graph G=(V,E) with V being the cities. Each spy i has a starting point sᵢ ∈V and needs to reach the consulate of a friendly nation; these consulates are in a known set of cities T⊆V. In order to move undetected, the spies agree that at most c of them should ever pass through any one city. Our goal is to find a set of paths, one for each of the spies (or detect that the requirements cannot be met). Model this problem as a flow network. Specify the vertices, edges and capacities, and explain how a maximum flow in your network can be transformed into a feasible solution to the original problem. You do not need to explain how to solve the max-flow instance itself.

Answer 1

To model the problem, we can define vertices for cities, spies, and the source and sink. Edges and capacities can represent the constraints and requirements. A maximum flow in the network can be transformed into a feasible solution for the spies.

Step-by-step explanation:

To model the given problem as a flow network, we can define the following:

Vertices:
- The set of vertices V will represent the cities in the railway system.
- For each spy, there will be two additional vertices, denoted as sᵢ_in and sᵢ_out, representing the start and end points of their path.
- There will be a source vertex S and a sink vertex T.

Edges and Capacities:
- For each city v∈V, there will be an edge (v_in, v_out) with capacity c, representing the constraint that at most c spies can pass through any city.
- For each spy i, there will be an edge (sᵢ_in, sᵢ_out) with capacity 1, representing the path of that spy.
- For each city v∈V and spy i, there will be an edge (v_out, sᵢ_in) with capacity 1, representing the connection between the city and the spy's path.
- For each city v∈T, there will be an edge (v_out, T) with capacity c, representing the requirement for spies to reach the consulate of a friendly nation.
- For each spy i, there will be an edge (sᵢ_out, T) with capacity 1, representing the requirement for each spy to reach the consulate.

A maximum flow in this network can be transformed into a feasible solution to the original problem by considering the flow on the paths of each spy. The flow will determine the cities that each spy passes through, and the constraints on the maximum number of spies passing through a city will be satisfied. If the maximum flow in the network saturates all the edges from the spies to the sink, it means that each spy successfully reaches the consulate of a friendly nation.