Online routing and searching on graphs with blocked edges

Abstract

We study routing and searching decisions of a search-and-detection (SDT) team on a road network under online uncertainty setting. Given an undirected edge-weighted bounded graph, a static target is positioned at an unknown vertex among potential target vertices in the graph. A non-negative search cost is given on each of the potential target vertices. The target is detected once the SDT searches its corresponding vertex. There may be some non-recoverable online blockages in the graph, but the existence of blockages is unknown to the SDT initially. If a blockage exists in the graph, it is disclosed online once the SDT visits one of its end-vertices. The graph stays connected when the blockages are omitted from it. The SDT begins from a certain vertex and aims to identify a route without any blocked edges which detects the target with minimum total traveling and search cost. We analyze this problem from a competitive analysis perspective under two scenarios with and without blockages. For the scenario with blockages, we provide a tight lower bound on the competitive ratio of deterministic solutions, an optimal deterministic solution, a randomized solution with a bounded expected competitive ratio, together with lower and upper bounds on the expected competitive ratio of the optimal randomized solutions. For the scenario without blockages, we provide tight lower bounds as well as optimal deterministic and randomized solutions.