Draw four locations on a page and connect them with roads. Give each road a travel cost. It seems like one mathematical object: a network of points and links. But ask three different questions about that drawing and you may need three quite different algorithms.
What is the cheapest way to reach a destination? How should several deliveries share limited road capacity? Which new connection would be worth building? The network is the same. The decisions are not.
First, find a route
Imagine a depot A and a destination D with two available corridors. The route through B costs four units in total; the route through C costs six. If one vehicle needs to reach D and all road costs are nonnegative, the answer is easy to understand: use B. Shortest-path algorithms can exploit this network structure to find an optimal path without enumerating every possible tour.
Now suppose the operation needs to move two vehicles at once. The final link through B can handle only one during the relevant period. Both vehicles choosing the individually shortest route would overload it. The second vehicle must use the longer corridor, or the operation must delay one movement. The cost of the two movements is then four plus six, not four plus four.
The road map has not changed. What changed is the mathematical question. Instead of finding one good path, we are assigning several flows that compete for shared capacity. Optimizing each trip independently does not solve the combined problem.
The cost of the two movements is then four plus six, not four plus four.
Then consider changing the network itself
Suppose the organization could upgrade that bottleneck link for eight units, allowing both vehicles to use the shorter corridor. Each subsequent two-vehicle operation would save two cost units under these simplified assumptions. If the same demand and prices recur, the upgrade would offset its eight-unit investment after four such operations. Whether that is worthwhile in reality depends on time horizon, financing, future demand and many other constraints, but the key idea is already visible.
Better solutions can mean less congestion, lower operating cost or better use of capacity.
This third question is network design. We are no longer choosing paths through an unchangeable graph. We can change the graph, at a cost, to improve future decisions. The benefit of a new edge depends on how traffic will be routed after it is built. Infrastructure and operation have become interdependent decisions.
Where the algorithms differ
Shortest paths have powerful specialized algorithms under familiar conditions. Multi-commodity routing with capacities adds interactions among flows. Network design often brings discrete construction choices and investment budgets into the model. Treating them all as “a graph problem” loses what makes each mathematically distinctive.
This is a useful lesson for telecommunications, transport, energy and computing infrastructure. A network diagram shows which entities are connected. An optimization model must show what may change, what may not, and what the organization is trying to improve. A good algorithm then uses that structure to avoid searching through possibilities that are irrelevant to the real decision.
Better solutions can mean less congestion, lower operating cost or better use of capacity. Faster solutions can make it possible to reconsider routes after a disruption, or compare investments before committing capital. The business benefit follows from the actual objective and the system's ability to put the plan into practice.