We study an optimal investment problem that arises in the context of the
vehicle-sharing system. Given a set of locations to build stations, we need to
determine i) the sequence of stations to be built and the number of vehicles to
acquire in order to obtain the target state where all stations are built, and
ii) the number of vehicles to acquire and their allocation in order to maximize
the total profit returned by operating the system when some or all stations are
open. The profitability associated with operating open stations, measured over
a specific time period, is represented as a linear optimization problem applied
to a collection of open stations. With operating capital, the owner of the
system can open new stations. This property introduces a set-dependent aspect
to the duration required for opening a new station, and the optimal investment
problem can be viewed as a variant of the Traveling Salesman Problem (TSP) with
set-dependent cost. We propose an A* search algorithm to address this
particular variant of the TSP. Computational experiments highlight the benefits
of the proposed algorithm in comparison to the widely recognized Dijkstra
algorithm and propose future research to explore new possibilities and
applications for both exact and approximate A* algorithms.

在车辆共享系统中，我们研究了一个最优投资问题，其中包括确定建立站点的顺序、车辆数量以及获得目标状态的策略，以及通过经营系统来最大化总利润的车辆数量和分配策略。该问题可以视为具有集合相关成本的旅行推销员问题的变体，并使用 A * 搜索算法进行解决。

车辆共享系统中一种优化投资问题的 A * 搜索算法

A* search algorithm for an optimal investment problem in vehicle-sharing  systems

Planning as theorem proving in situation calculus was abandoned 50 years ago
as an impossible project. But we have developed a Theorem Proving Lifted
Heuristic (TPLH) planner that searches for a plan in a tree of situations using
the A* search algorithm. It is controlled by a delete relaxation-based domain
independent heuristic. We compare TPLH with Fast Downward (FD) and Best First
Width Search (BFWS) planners over several standard benchmarks. Since our
implementation of the heuristic function is not optimized, TPLH is slower than
FD and BFWS. But it computes shorter plans, and it explores fewer states. We
discuss previous research on planning within KR\&R and identify related
directions. Thus, we show that deductive lifted heuristic planning in situation
calculus is actually doable.

本研究介绍了一种利用启发式算法解决定理证明和计划制定问题的方法，并将其应用于 situation calculus 中。该方法通过使用 A * 搜索算法排列一系列情境，并利用删除松弛法控制启发式规划器，获得较短的计划方案，并探索较少的状态。实验表明，该方法可以应用于较大规模的问题。

启发式规划的定理证明

Planning as Theorem Proving with Heuristics

Path finding in graphs is one of the most studied classes of problems in
computer science. In this context, search algorithms are often extended with
heuristics for a more efficient search of target nodes. In this work we combine
recent advancements in Neural Algorithmic Reasoning to learn efficient
heuristic functions for path finding problems on graphs. At training time, we
exploit multi-task learning to learn jointly the Dijkstra's algorithm and a
consistent heuristic function for the A* search algorithm. At inference time,
we plug our learnt heuristics into the A* algorithm. Results show that running
A* over the learnt heuristics value can greatly speed up target node searching
compared to Dijkstra, while still finding minimal-cost paths.

本文利用神经演算法推导出高效的启发式路径搜索算法，结合 Dijkstra 算法和 A * 算法中的一致的启发函数，并将其推广应用于图形中的路径搜索问题，结果表明相较于 Dijkstra 算法，应用基于学习到的 A * 算法中的启发函数的改进搜索在无损质量的情况下，实现了大幅的速度提升。