AI 中文总结
针对NP难的多车场带容量约束车辆路径问题，提出Cluster-First和Match-First两种基于图匹配的算法，可在多项式时间内求解小规模实例，在大规模实例上速度远超基准且路径质量相当，还支持新增客户时低成本重规划。
AI 中文摘要
多车场带容量约束车辆路径问题（MDCVRP）要求从多个带容量约束的车场出发，为一组客户规划总成本最低的配送路线。与大多数车辆路径问题的变体一样，该问题属于NP难问题，因此实用求解器必须在解的质量与求解速度之间进行权衡。我们从图匹配的视角重新审视这一权衡，对最初为旅行锦标赛问题开发的匹配式构造方法进行适配，提出了Cluster-First和Match-First两种算法，将路径规划问题转化为一系列最小权匹配问题。这并非简单的启发式方法：我们证明，对于最多包含两个目标点的路径，该匹配公式可在多项式时间内精确求解任意数量车场的MDCVRP；在结构化场景下，两种算法均为常数因子近似算法，且紧因子为2。该匹配最优解与精确组合拍卖最优解一致，因此拍卖可作为可靠的质量基准。在包含1000个客户和20个车场的实例上，我们的方法在路径长度上与该基准相当或略优，同时运行速度快2至3个数量级——耗时仅数十毫秒，而基准方法需数十秒，在该规模下精确求解器和基于拍卖的求解器已无法实用。由于Cluster-First算法独立规划每个车场的路径，当新增客户时，该方法还能以低成本重新规划路径。
英文摘要
The Multi-Depot Capacitated Vehicle Routing Problem (MDCVRP) asks for minimum-cost delivery tours from several capacitated depots to a set of customers. Like most vehicle-routing variants it is NP-hard, so practical solvers must trade solution quality against speed. We revisit this trade-off through the lens of graph matching. Adapting a matching-based construction first developed for the Traveling Tournament Problem, we present two algorithms, Cluster-First and Match-First, that reduce routing to a sequence of minimum-weight matchings. This is more than a heuristic. We prove that for tours of up to two targets the matching formulation solves the MDCVRP exactly in polynomial time for any number of depots, and that both algorithms are constant-factor approximations, with a tight factor of two, in the structured regimes. This matching optimum coincides with the exact combinatorial-auction optimum, so the auction serves as a strong quality baseline. On instances of 1000 customers and 20 depots our methods match or slightly beat that baseline in tour length while running two to three orders of magnitude faster, in tens of milliseconds against tens of seconds, a scale at which exact and auction-based solvers become impractical. Because Cluster-First routes each depot independently, the approach also re-routes cheaply when new customers arrive.
Comments19 pages, 5 figures, 1 table
