🤖 AI Summary
This paper studies the Weighted Expansion Search problem on graphs: a searcher starts from a designated vertex and iteratively expands along edges to unvisited vertices, with edge lengths representing traversal time; the objective is to minimize the weighted sum of vertex weights multiplied by their first-visit times. As an NP-hard problem, it had long been stuck at an 8-approximation barrier. We break this barrier via a unified framework integrating greedy construction, randomized rounding, Euclidean geometric partitioning, and dynamic programming, complemented by hardness analysis under exponential-time hypotheses. Our contributions are the first: (i) a $(2e+varepsilon)$-approximation algorithm for general graphs; (ii) a $2e$-approximation algorithm for unit-weight graphs; and (iii) a PTAS for Euclidean graphs (for any $varepsilon > 0$). These results substantially improve approximation ratios and provide the strongest known theoretical guarantees for all three graph classes.
📝 Abstract
A searcher faces a graph with edge lengths and vertex weights, initially having explored only a given starting vertex. In each step, the searcher adds an edge to the solution that connects an unexplored vertex to an explored vertex. This requires an amount of time equal to the edge length. The goal is to minimize the weighted sum of the exploration times over all vertices. We show that this problem is hard to approximate and provide algorithms with improved approximation guarantees. For the general case, we give a $(2mathrm{e}+varepsilon)$-approximation for any $varepsilon>0$. For the case that all vertices have unit weight, we provide a $2mathrm{e}$-approximation. Finally, we provide a PTAS for the case of a Euclidean graph. Previously, for all cases only an $8$-approximation was known.
