On the tree search problem with non-uniform costs

Abstract

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 NPcomplete 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. TCS 2012]. 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(log n log log n)-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.

Edge and Pair Queries -- Random Graphs and Complexity

“…It is natural to further generalize edge queries to the weighted variant, where the weight of an edge represent the duration of a query [28]. Such an extension turns out to be N P-complete for some quite narrow subclasses of trees [12,13,15]. On the other hand, there has been a series of results on approximation algorithms [12,13,15] with the best approximation ratio to date being O( √ log n) that was proved in [16].…”

“…Such an extension turns out to be N P-complete for some quite narrow subclasses of trees [12,13,15]. On the other hand, there has been a series of results on approximation algorithms [12,13,15] with the best approximation ratio to date being O( √ log n) that was proved in [16]. Finally, let us mention another generalization of binary search in linear orders to graphs [22,18] with some interesting applications in machine learning [21].…”

The Complexity of Bicriteria Tree-Depth

“…The weights naturally represent the number of time units required for performing particular operation (in the earlier mentioned applications, we took a silent assumption that all operations have unit duration). Once we introduce the weights, the problem becomes strongly N P-hard even for some restricted classes of trees [5,6,8]. For algorithmic results on weighted paths see e.g.…”

“…Note that even such a simple structure as paths is practically interesting in this context since our problem generalizes binary search. For weighted trees, there is a number of works improving on possible approximation ratio achievable in polynomial-time [5,6,8], with the best one to date having an approximation factor of O( √ log n) [10], and it is unknown whether this is best possible-in particular, a challenging open question regarding this line of research is whether a constant-factor approximation is feasible for weighted trees. Also, many intriguing problems remain open in the area of online competitiveness that has been considered, e.g., for line graphs of specific star-like graphs [3] or just for vertex variant on trees [22].…”

Approximation Strategies for Generalized Binary Search in Weighted Trees

“…The problem of computational complexity for weighted trees attracted a lot of attention. On the negative side, it has been proved that it is strongly NP-hard to compute an optimal search strategy [8] for bounded diameter trees, which has been improved by showing hardness for several specific topologies: trees of diameter at most 6, trees of degree at most 3 [6] and spiders [7] (trees having at most one node of degree greater than two). On the other hand, polynomial-time algorithms exist for weighted trees of diameter at most 5 and weighted paths [6].…”