Combinatorial bilevel congestion pricing (CBCP), a variant of the discrete network design problem, seeks to minimize the total travel time experienced by all travelers in a road network, by strategically selecting toll locations and determining the corresponding charges. Conventional wisdom suggests that these problems are intractable since they have to be formulated and solved with a significant number of integer variables. Here, we devise a scalable local algorithm for the CBCP problem that guarantees convergence to a Kuhn-Tucker-Karush point. Our approach is novel in that it eliminates the use of integer variables altogether, instead introducing a cardinality constraint that limits the number of toll locations to a user-specified upper bound. The resulting bilevel program with the cardinality constraint is then transformed into a block-separable, single-level optimization problem that can be solved efficiently after penalization and decomposition. We are able to apply the algorithm to solve, in about 20 minutes, a CBCP instance with up to 3,000 links, of which hundreds can be tolled. To the best of our knowledge, no existing algorithm can solve CBCP problems at such a scale while providing any assurance of convergence.

翻译：组合双层拥堵定价（CBCP）是离散网络设计问题的一个变体，旨在通过策略性地选择收费路段位置并确定相应收费金额，最小化道路网络中所有出行者的总出行时间。传统观点认为此类问题难以处理，因为其建模与求解需要大量整数变量。本文为CBCP问题设计了一种可扩展的局部算法，该算法能保证收敛至Kuhn-Tucker-Karush点。我们的方法创新之处在于完全避免了整数变量的使用，转而引入基数约束，将收费路段数量限制在用户指定的上限内。随后，将带有基数约束的双层规划问题转化为块可分离的单层优化问题，该问题经过惩罚化和分解处理后即可高效求解。该算法能在约20分钟内求解包含多达3000条路段（其中数百条可设为收费路段）的CBCP实例。据我们所知，目前尚无其他算法能在保证收敛性的前提下处理如此规模的CBCP问题。