Searching in trees with $k$-up-modular cost functions

Consider the following generalization of the classic binary search problem: a searcher is required to find a hidden vertex $x$ in a tree $T$. To do so, they iteratively perform queries to an oracle, each 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 connected component of $T-v$ which contains $x$. Additionally, each vertex $v$ may have a different query cost $c(v)$. The goal is to find the optimal querying strategy which minimizes the worst case 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 general trees. Following the recent studies of [Dereniowski et al. 2022, Dereniowski et al. 2024], instead of restricted classes of trees, we explore restrictions on the cost function. We generalize the notion of up-monotonic functions and introduce the concept of \textit{$k$-up-modularity}. We show that an $O(\log\log n)$-approximate solution can be found within $k^{O(\log k)}\cdot\text{poly}(n)$ time.