Bilevel Optimization for Traffic Mitigation in Optimal Transport Networks

TL;DR

This work develops adaptation rules that leverage optimal transport theory to effectively route passengers along their shortest paths while also strategically tuning edge weights to optimize traffic, and proves the efficacy of this approach on synthetic networks and on real data.

Abstract

Global infrastructure robustness and local transport efficiency are critical requirements for transportation networks. However, since passengers often travel greedily to maximize their own benefit and trigger traffic jams, overall transportation performance can be heavily disrupted. We develop adaptation rules that leverage Optimal Transport theory to effectively route passengers along their shortest paths while also strategically tuning edge weights to optimize traffic. As a result, we enforce both global and local optimality of transport. We prove the efficacy of our approach on synthetic networks and on real data. Our findings on the International European highways suggest that thoughtfully devised routing schemes might help to lower car-produced carbon emissions.

Figures (12)

• Figure 1: Bilevel optimization scheme on a lattice. Entry and exit inflows are the red and blue nodes, respectively. Initially, (green) fluxes distribute minimizing the travel cost $w_e(t=0) = \ell_e$, being the length of an edge. If they exceed $\theta$ they get penalized, hence, the network manager tunes the weights to encourage rerouting over more expensive (red), or cheaper (blue) edges (for a companion Fig. si).
• Figure 2: Overview of the routing schemes. (a) $J$ and $\Omega$ against $\theta$. (b) Trade-off $J - J_\mathrm{OT}$ vs. $\Omega - \Omega_0$ with varying $(\theta, q,\xi)$. Non-dominated points for $\theta / \theta^{\star} \simeq \{ 0.06, 0.2, 0.3,0.4 \}$ are in red. (c) BROT's networks at different $\theta$. Edge widths are proportional to the average fluxes in $50$ runs of the algorithm. Gray edge contours are fluxes' standard deviations. (d) Cost (left) and flux (right) networks for all methods and $\theta/\theta^\star=0.4$. Flux networks are as in (c), whereas edges in the cost networks are colored with $\rho$ and their widths are proportional to the fluxes. The black rectangle frames a region where the network manager triggers high congestion. We conveniently normalize $\theta^{\star}$ and $\rho$.
• Figure 3: Measuring traffic congestion, $D=8$. (a) Gini coefficient against $\theta$. (b) $T_\theta(s)$ against $\theta$. Solid lines correspond to low sensitivity $s=1$ and dashed ones to $s=50$, in red we draw $T_\infty$ (free flow). Shades are standard deviations over $50$ realizations of the algorithms.
• Figure 4: E-road transport networks. Nodes in red are 15 main cities taken as passenger inflows, their size is proportional to the entry inflows. Edge widths are the total number of passengers $\sum_i |\tilde{F}_e^i|$, gray shades are standard deviations over $50$ realizations of the algorithms.
• Figure S1: Bilevel optimization scheme in detail. We plot costs and network topology for different adaptation rules. Histograms and networks labeled with purple, blue, orange, and green squares correspond to coordinated traffic, OT, PSGD, and BROT, respectively. Hatch styles are $J$ and $\Omega$. Edges in black are proportional to $\sum_i |F_e^i|$, while those colored with $\rho$ express the change in (the normalized) $w$.