Collectively optimal routing for congested traffic limited by link capacity

We show that the capacity of a complex network that models a city street grid to support congested traffic can be optimized by using routes that collectively minimize the maximum ratio of betweenness to capacity in any link. Networks with a heterogeneous distribution of link capacities and with a heterogeneous transport load are considered. We find that overall traffic congestion and average travel times can be significantly reduced by a judicious use of slower, smaller capacity links.

💡 Research Summary

The paper tackles the problem of traffic congestion in urban street networks by proposing a routing strategy that minimizes the worst‑case ratio of link betweenness to link capacity. The authors model a city’s road system as a complex graph (G(V,E)) where each edge ((i,j)) carries a physical capacity (C_{ij}) (vehicles per unit time) and a traffic load measured by betweenness (B_{ij}), i.e., the number of shortest‑path routes that traverse the link. Traditional routing schemes that rely solely on shortest‑path distances tend to concentrate traffic on a few high‑capacity arteries, causing those links to become bottlenecks while under‑utilizing smaller streets.

To avoid this imbalance, the authors introduce the metric (R_{ij}=B_{ij}/C_{ij}) and formulate the routing problem as the minimization of (\max_{(i,j)} R_{ij}). This objective forces the traffic distribution to be as uniform as possible relative to each link’s capacity, ensuring that no single link operates near saturation while others remain idle.

The solution methodology consists of two main stages. First, an initial traffic assignment is generated using conventional shortest‑path routing, allowing the computation of the baseline betweenness values for all links. Second, the routing is iteratively refined by introducing a Lagrange multiplier (\lambda) that represents a candidate upper bound on the ratio (R_{ij}). For a given (\lambda), the authors define a modified edge weight

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