We propose a combinatorial optimisation model called Limited Query Graph Connectivity Test. We consider a graph whose edges have two possible states (on/off). The edges' states are hidden initially. We could query an edge to reveal its state. Given a source s and a destination t, we aim to test s-t connectivity by identifying either a path (consisting of only on edges) or a cut (consisting of only off edges). We are limited to B queries, after which we stop regardless of whether graph connectivity is established. We aim to design a query policy that minimizes the expected number of queries. If we remove the query limit B (i.e., by setting B to the total number of edges), then our problem becomes a special case of (monotone) Stochastic Boolean Function Evaluation (SBFE). There are two existing exact algorithms that are prohibitively expensive. They have best known upper bounds of O(3^m) and O(2^{2^k}) respectively, where m is the number of edges and k is the number of paths/cuts. These algorithms do not scale well in practice. We propose a significantly more scalable exact algorithm. Our exact algorithm works by iteratively improving the performance lower bound until the lower bound becomes achievable. Even when our exact algorithm does not scale, it can be used as an anytime algorithm for calculating lower bound. We experiment on a wide range of practical graphs. We observe that even for large graphs (i.e., tens of thousands of edges), it mostly takes only a few queries to reach conclusion, which is the practical motivation behind the query limit B. B is also an algorithm parameter that controls scalability. For small B, our exact algorithm scales well. For large B, our exact algorithm can be converted to a heuristic (i.e., always pretend that there are only 5 queries left). Our heuristic outperforms all existing heuristics ported from SBFE and related literature.

翻译：我们提出一种名为有限查询图连通性测试的组合优化模型。考虑一个图，其边具有两种可能状态（开/关），且初始状态未知。我们可以通过查询一条边来揭示其状态。给定源点s和汇点t，我们旨在通过识别一条由开边组成的路径或由关边组成的割集来测试s-t连通性。查询次数被限制为B次，无论是否建立图连通性，我们都将在B次查询后停止。我们的目标是设计一个使期望查询次数最小化的查询策略。若移除查询限制B（即令B等于总边数），则问题退化为（单调）随机布尔函数评估（SBFE）的特例。现有两种精确算法但因复杂度极高而难以实用——其已知最优上界分别为O(3^m)和O(2^{2^k})，其中m为边数，k为路径/割集数量。这些算法在实际中扩展性较差。我们提出了一种扩展性显著提升的精确算法：通过迭代改进性能下界直至其可实际达成。即使该精确算法无法完全扩展，也可作为随时算法用于计算下界。我们在多种实际图上开展实验，观察到对于大规模图（如数万条边），通常仅需少量查询即可得出结论——这正是查询限制B的实际动机。B还是控制算法扩展性的参数：对于小B，我们的精确算法扩展性良好；对于大B，精确算法可转换为启发式方法（即始终假设仅剩5次查询）。该启发式方法在性能上超越了所有从SBFE及相关文献移植的现有启发式方法。