We introduce a search problem generalizing the typical setting of Binary Search on the line. Similar to the setting for Binary Search, a target is chosen adversarially on the line, and in response to a query, the algorithm learns whether the query was correct, too high, or too low. Different from the Binary Search setting, the cost of a query is a monotone non-decreasing function of the distance between the query and the correct answer; different functions can be used for queries that are too high vs. those that are too low. The algorithm's goal is to identify an adversarially chosen target with minimum total cost. Note that the algorithm does not even know the cost it incurred until the end, when the target is revealed. This abstraction captures many natural settings in which a principal experiments by setting a quantity (such as an item price, bandwidth, tax rate, medicine dosage, etc.) where the cost or regret increases the further the chosen setting is from the optimal one. First, we show that for arbitrary symmetric cost functions (i.e., overshooting vs. undershooting by the same amount leads to the same cost), the standard Binary Search algorithm is a 4-approximation. We then show that when the cost functions are bounded-degree polynomials of the distance, the problem can be solved optimally using Dynamic Programming; this relies on a careful encoding of the combined cost of past queries (which, recall, will only be revealed in the future). We then generalize the setting to finding a node on a tree; here, the response to a query is the direction on the tree in which the target is located, and the cost is increasing in the distance on the tree from the query to the target. Using the k-cut search tree framework of Berendsohn and Kozma and the ideas we developed for the case of the line, we give a PTAS when the cost function is a bounded-degree polynomial.

翻译：我们引入了一个搜索问题，该问题推广了直线上二分搜索的经典设定。与二分搜索设定类似，对手在直线上选择目标点，算法根据查询结果得知该查询是否正确、过高或过低。与二分搜索设定不同之处在于，查询的代价是查询点到目标点距离的单调非递减函数；针对过高与过低的查询可采用不同代价函数。算法的目标是以最小总代价识别出对手选择的目标点。注意，算法直到目标点暴露时才知道已产生的实际代价。这一抽象模型涵盖了许多自然场景，其中决策者通过设置某个量（如商品价格、带宽、税率、药物剂量等）进行实验，而设置量偏离最优值越远，其代价或遗憾越大。首先，我们证明对于任意对称代价函数（即相同幅度的高估与低估产生相同代价），标准二分搜索算法具有4倍近似比。随后证明当代价函数为有界次数的距离多项式时，可通过动态规划最优求解该问题；这依赖于对历史查询组合代价（注：这些代价将仅在后续时刻被揭示）的精心编码。最后我们将设定推广至树上节点搜索：此时查询的响应结果为目标所在树上的方向，代价随查询点到目标点的树上距离递增。基于Berendsohn与Kozma提出的k-割搜索树框架，以及我们针对直线情形发展的技术，当代价函数为有界次数多项式时，我们给出了一个多项式时间近似方案(PTAS)。