Maximum Covering Network Design on Graphs with Low Connectivity: Dynamic Programming and Block-Cut Trees

Abstract

Planning accessible public services such as health care, emergency response, and schools often requires not only choosing where to open facilities but also improving the network that connects people to them, for example upgrading flood-prone roads in vulnerable regions. Most location models, however, take the network as fixed and the budget as given. We study the Maximum Covering Network Design Problem, in which a single budget is shared between opening facilities and upgrading weak links to maximize the population within a target travel distance of an open facility. The problem is hard even on the simplest networks, and planners usually want to see how coverage grows with the budget, not a single plan. We develop an exact dynamic-programming framework that exploits a property common to real road networks: their low connectivity, with many cut points whose removal disconnects the network. On trees, the recursion is self-contained: each state reduces to a few simple facility and upgrade choices that are fast to compute without a solver, giving predictable running times; for larger budgets and travel distances it outperforms solving the MILP formulation directly. On general low-connectivity networks, the framework decomposes the problem at the cut points and embeds a given MILP formulation to solve the resulting pieces, coordinating them through coverage conditions at the interfaces. This lets us compare a formulation on its own against the same formulation inside the framework: across 306 test cases the framework matches or outperforms direct solving on more than 80% of instances. Because it evaluates all budget levels in a single run, it also yields the full coverage-versus-budget curve at no extra cost, whereas direct solving must split its time across individual budgets.