Searching in trees with k-up-modular weight functions

We consider the following generalization of the binary search problem: A searcher is required to find a hidden element $x$ in a tree $T$. To do so, they iteratively perform queries to an oracle about a chosen vertex $v$. After each such call, the oracle responds whether the target was found and if not, the searcher receives as a reply the neighbor of $v$ that lays on the shortest path towards $x$. Additionally, each vertex $v$ may have a different query cost $w(v)$. The goal is to find the optimal querying strategy for the searcher which minimizes the worst case query cost required to find $x$. The problem is known to be NP-hard even in restricted classes of trees such as bounded diameter spiders [Cicalese et al. 2016] and no constant factor approximation algorithm is known for the general case. Inspired by recent studies [Dereniowski et al. 2022, Dereniowski et al. 2024], instead of restricted classes of trees, we explore restrictions on the weight function. We introduce the concept of a heavy group set of a vertex $HG(v,w)$. We show that if for every $v\in T$: $|HG\br{v,w}|\leq k$ an $O(\log\log n)$-approximation can be found within $2^{O(\log^2k)}\cdot\text{poly}(n)$ time.