Many interesting search problems can be formulated as bi-objective search problems, that is, search problems where two kinds of costs have to be minimized, for example, travel distance and time for transportation problems. Instead of looking for a single optimal path, we compute a Pareto-optimal frontier in bi-objective search, which is a set of paths in which no two paths dominate each other. Bi-objective search algorithms perform dominance checks each time a new path is discovered. Thus, the efficiency of these checks is key to performance. In this article, we propose algorithms for two kinds of bi-objective search problems. First, we consider the problem of computing the Pareto-optimal frontier of the paths that connect a given start state with a given goal state. We propose Bi-Objective A* (BOA*), a heuristic search algorithm based on A*, for this problem. Second, we consider the problem of computing one Pareto-optimal frontier for each state s of the search graph, which contains the paths that connect a given start state with s. We propose Bi-Objective Dijkstra (BOD), which is based on BOA*, for this problem. A common feature of BOA* and BOD is that all dominance checks are performed in constant time, unlike the dominance checks of previous algorithms. We show in our experimental evaluation that both BOA* and BOD are substantially faster than state-of-the-art bi-objective search algorithms.

许多有趣的搜索问题可以表述为双目标搜索问题，即必须最小化两种成本的搜索问题，例如交通问题的旅行距离和时间。我们不是寻找单一的最优路径，而是计算双目标搜索中的帕累托最优边界，这是一组路径，其中没有两条路径相互支配。每次发现新路径时，双目标搜索算法都会执行优势检查。因此，这些检查的效率是性能的关键。在本文中，我们提出了两种双目标搜索问题的算法。首先，我们考虑计算连接给定起始状态和给定目标状态的路径的帕累托最优边界的问题。我们提出了基于 A* 的启发式搜索算法 Bi-Objective A* (BOA*)，对于这个问题。其次，我们考虑为每个状态计算一个帕累托最优边界的问题搜索图的 s ，其中包含将给定起始状态与s连接的路径。针对这个问题，我们提出了基于 BOA* 的 Bi-Objective Dijkstra (BOD)。BOA* 和 BOD 的一个共同特点是所有优势检查都是在恒定时间内执行的，这与以前算法的优势检查不同。我们在实验评估中表明，BOA* 和 BOD 都比最先进的双目标搜索算法快得多。