Improved Approximation Algorithms for the Average-Case Tree Searching Problem

Abstract

We study the following tree search problem: in a given tree T = (V , E) a vertex has been marked and we want to identify it. In order to locate the marked vertex, we can use edge queries. An edge query e asks in which of the two connected components of T \ e the marked vertex lies. The worst-case scenario where one is interested in minimizing the maximum number of queries is well understood, and linear time algorithms are known for finding an optimal search strategy. Here we study the more involved average-case analysis: A function w : V → R+ is given which measures the likelihood for a vertex to be the one marked, and we seek to determine the strategy (decision tree) that minimizes the weighted average number of queries. In a companion paper we prove that the above tree search problem is NP- complete even for the class of trees of bounded diameter or bounded degree. Here, we match this complexity result with a tight algorithmic analysis of the bounded degree instances. We show that any optimal strategy (i.e., one that minimizes the weighted average number of queries) performs at most O(Delta(T )(log |V | + log(w(T )/wmin))) queries in the worst case, where w(T ) is the sum of the likelihoods of the vertices of T , wmin is the minimum positive likelihood over the vertices of T and Delta(T ) is the maximum degree of T . We combine this result with a non-trivial exponential time algorithm to provide an FPTAS for trees with bounded degree. We also show that for unbounded instances a natural greedy strategy attains a 1.62-approximation, improv- ing upon the best known 14-approximation guarantee, previously provided by two of the authors

Cost-Effective Algorithms for Average-Case Interactive Graph Search

“…Next, we consider two special cases of AIGS: (i) the input hierarchy is a tree, and (ii) the probability of every node being the target is identical, i.e., p(v) = 1/n for every v ∈ V . We show that the approximation ratios achieved by the greedy policy under the two scenarios are much better than O(log n) leveraging the theoretical developments of poset [9] and decision tree [30].…”

“…Let D be the decision tree constructed by greedy policy and D * be the optimal decision tree, with respect to the cost function defined in (2). For such a tree-like poset, Cicalese et al [9] proved that the cost of D satisfies…”

“…In fact, it has been proved that there is no o(log n)-approximate algorithm unless NP ⊆ DTIME n O(log log n) [8]. However, surprisingly, the simple greedy method can provide a constant approximate guarantee for the tree-like poset [9,27].…”

Noisy Interactive Graph Search

“…2 and there exists a fully polynomial time approximation scheme based on dynamic programming on a tree hierarchy [7]; while on a DAG hierarchy, the rounded greedy algorithm can achieve an approximation ratio of 𝑂 (log(𝑛)) [4] and no ratios better than 𝑜 (log(𝑛)) can be achieved unless P = NP [6]. Again, noise is not considered in these results.…”

Competitive Online Search Trees on Trees

“…Our definition of search trees is based on vertex queries: every node of the search tree is associated with a vertex of the searched graph. In contrast, edge queries have also been studied [BFN99, OP06, MOW08, CKL + 16, CJLM11,CJLM14], in which every node of the search tree is associated with an edge of the graph, and the oracle points to one of the two connected components containing the sought vertex. is is sometimes cast as searching in a partial order.…”