Approximating the Average-Case Graph Search Problem with Non-Uniform Costs
Michał Szyfelbein
TL;DR
The paper studies the Graph Search Problem (GSP), where a hidden target vertex $x$ in a graph $G$ is found via adaptive vertex queries with nonuniform costs $c(v)$ and weights $w(v)$. It exploits a connection to vertex separators to design efficient approximation algorithms: a $(4+\epsilon)$-approximation for trees achieved by a pseudo-exact Weighted $\alpha$-Separator FPTAS, and an $O(\sqrt{\log n})$-approximation for general graphs via a Min-Ratio Vertex Cut framework. These results provide the first approximation guarantees for the vertex-query, nonuniform-cost/weight variant of GSP, with NP-hardness established for even restricted cases and detailed discussions in the appendices. The approaches are conceptually simple and practically appealing, suggesting avenues for SDP-based relaxations and tighter bounds in future work.
Abstract
Consider the following generalization of the classic binary search problem: A searcher is required to find a hidden target vertex $x$ in a graph $G$. 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 in $G-v$ which contains $x$. Additionally, each vertex $v$ may have a different query cost $c(v)$ and a different weight $w(v)$. The goal is to find the optimal querying strategy which minimizes the weighted average-case cost required to find $x$. The problem is NP-hard even for uniform weights and query costs. Inspired by the progress on the edge query variant of the problem [SODA '17], we establish a connection between searching and vertex separation. By doing so, we provide an $O(\sqrt{\log n})$-approximation algorithm for general graphs and a $(4+ε)$-approximation algorithm for the case when the input is a tree.