On the tree search problem with non-uniform costs

Searching in partially ordered structures has been considered in the context of information retrieval and efficient tree-like indices, as well as in hierarchy based knowledge representation. In this paper we focus on tree-like partial orders and consider the problem of identifying an initially unknown vertex in a tree by asking edge queries: an edge query e returns the component of T−e containing the vertex sought for, while incurring some known cost c(e). The Tree Search Problem with Non-Uniform Cost is the following: given a tree T on n vertices, each edge having an associated cost, construct a strategy that minimizes the total cost of the identification in the worst case. Finding the strategy guaranteeing the minimum possible cost is an NP-complete problem already for input trees of degree 3 or diameter 6. The best known approximation guarantee was an O(log⁡n/log⁡log⁡log⁡n)-approximation algorithm of Cicalese et al. (2012) [4]. We improve upon the above results both from the algorithmic and the computational complexity point of view: We provide a novel algorithm that provides an O()-approximation of the cost of the optimal strategy. In addition, we show that finding an optimal strategy is NP-hard even when the input tree is a spider of diameter 6, i.e., at most one vertex has degree larger than 2.