On the Complexity of Searching in Trees: Average-case Minimization

TL;DR

This paper investigates the average-case complexity of searching in trees with weighted nodes, proving NP-completeness for certain classes, and providing algorithms with approximation guarantees and an FPTAS for bounded degree trees.

Contribution

It establishes the NP-completeness of the tree search problem for trees with diameter at most 4 and maximum degree 16, and offers approximation algorithms and an FPTAS for special cases.

Findings

NP-complete for diameter ≤ 4 and degree ≤ 16

Polynomial algorithm for diameter ≤ 3

Greedy algorithm achieves 2-approximation

Abstract

We focus on the average-case analysis: A function w : V -> Z+ is given which defines the likelihood for a node to be the one marked, and we want the strategy that minimizes the expected number of queries. Prior to this paper, very little was known about this natural question and the complexity of the problem had remained so far an open question. We close this question and prove that the above tree search problem is NP-complete even for the class of trees with diameter at most 4. This results in a complete characterization of the complexity of the problem with respect to the diameter size. In fact, for diameter not larger than 3 the problem can be shown to be polynomially solvable using a dynamic programming approach. In addition we prove that the problem is NP-complete even for the class of trees of maximum degree at most 16. To the best of our knowledge, the only known result in this direction is that the tree search problem is solvable in O(|V| log|V|) time for trees with degree at most 2 (paths). We match the above complexity results with a tight algorithmic analysis. We first show that a natural greedy algorithm attains a 2-approximation. Furthermore, for the bounded degree instances, we show that any optimal strategy (i.e., one that minimizes the expected number of queries) performs at most O(\Delta(T) (log |V| + log w(T))) queries in the worst case, where w(T) is the sum of the likelihoods of the nodes 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.