Limited Query Graph Connectivity Test
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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.
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