Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every planar graph G there is a graph H with treewidth at most 8 and a path P such that G ⊆ H P . We improve this result by replacing "treewidth at most 8" by "simple treewidth at most 6".

Dujmović, Joret, Micek, Morin, Ueckerdt and Wood [J. ACM 2020] proved that for every planar graph $G$ there is a graph $H$ with treewidth at most 8 and a path $P$ such that $G\subseteq H\boxtimes P$. We improve this result by replacing "treewidth at most 8" by "simple treewidth at most 6".

This paper studies the distance-based congestion pricing in a network considering the day-to-day dynamic traffic flow evolution process. It is well known that, after an implementation or adjustment of a new congestion toll scheme, the network environment will change and traffic flows will be nonequilibrium in the following days; thus it is not suitable to take the equilibrium-based indexes as the objective of the congestion toll. In the context of nonequilibrium state, prior research proposed a mini–max regret model to solve the distance-based congestion pricing problem in a network considering day-to-day dynamics. However, it is computationally demanding due to the calculation of minimal total travel cost for each day among the whole planning horizon. Therefore, in order to overcome the expensive computational burden problem and make the robust toll scheme more practical, we propose a new robust optimization model in this paper. The essence of this model, which is an extension of our prior work, is to optimize the worst condition among the whole planning period and ameliorate severe traffic congestions in some bad days. Firstly, a piecewise linear function is adopted to formulate the nonlinear distance toll, which can be encapsulated to a day-to-day dynamics context. A very clear and concise model named logit-type Markov adaptive learning model is then proposed to depict commuters’ day-to-day route choice behaviors. Finally, a robust optimization model which minimizes the maximum total travel cost among the whole planning horizon is formulated and a modified artificial bee colony algorithm is developed for the robust optimization model.