Given a hypergraph with uncertain node weights following known probability distributions, we study the problem of querying as few nodes as possible until the identity of a node with minimum weight can be determined for each hyperedge. Querying a node has a cost and reveals the precise weight of the node, drawn from the given probability distribution. Using competitive analysis, we compare the expected query cost of an algorithm with the expected cost of an optimal query set for the given instance. For the general case, we give a polynomial-time $f(\alpha)$-competitive algorithm, where $f(\alpha)\in [1.618+\epsilon,2]$ depends on the approximation ratio $\alpha$ for an underlying vertex cover problem. We also show that no algorithm using a similar approach can be better than $1.5$-competitive. Furthermore, we give polynomial-time $4/3$-competitive algorithms for bipartite graphs with arbitrary query costs and for hypergraphs with a single hyperedge and uniform query costs, with matching lower bounds.

翻译：根据已知概率分布的不确定节点重量,我们研究在确定每个顶端最小重量的节点身份之前尽可能多地询问几个节点的问题。 查询节点有成本, 并揭示从给定概率分布中得出的节点的精确重量。 我们通过竞争分析, 比较算法的预期查询成本和为给定点设定的最佳查询设定的预期成本。 在一般情况下, 我们给出一个多边- 时间 $f( ALpha) $- 有竞争力的算法, 其中, $f( ALpha)\ in [ 1.618 ⁇ epsilon, 2]$ 取决于一个基本脊椎盖问题的近似比率 $\ alpha$。 我们还表明, 使用类似方法的算法不能比1.5美元- 有竞争力强。 此外, 我们给具有任意查询成本的双面图和具有单一高端和统一查询成本的超直线算法, 与较低边框相匹配。