Summary: The modern warehouse is partially automated by robots. Instead of letting human workers walk into shelfs and pick up the required stock, big groups of autonomous mobile robots transport the inventory to the workers. Typically, these robots have an electric drive and need to recharge frequently during the day. When we scale this approach up, it is essential to place recharging stations strategically and as soon as needed so that all robots can survive. In this work, we represent a warehouse topology by a graph and address this challenge with the Online Connected Dominating Set problem (OCDS), an online variant of the classical Connected Dominating Set problem [S. Guha and S. Khuller, Algorithmica 20, No. 4, 374–387 (1998; Zbl 0895.68106)]. We are given an undirected connected graph \(G=(V, E)\) and a sequence of subsets of \(V\) arriving over time. The goal is to grow a connected subgraph that dominates all arriving nodes and contains as few nodes as possible. We propose an \(\mathcal{O}(\log^2 n)\)-competitive randomized algorithm for OCDS in general graphs, where \(n\) is the number of nodes in the input graph. This is the best one can achieve due to Korman’s randomized lower bound of \(\Omega(\log n\log m)\) [S. Korman, On the use of randomization in the online set cover problem. Rechovot: Weizmann Institute of Science (MSc Thesis) (2005)] for the related Online Set Cover problem [N. Alon et al., in: Proceedings of the thirty-fifth annual ACM symposium on theory of computing, STOC 2003. New York, NY: ACM Press. 100–105 (2003; Zbl 1192.90154)], where \(n\) is the number of elements and \(m\) is the number of subsets. We also run extensive simulations to show that our algorithm performs well in a simulated warehouse, where the topology of a warehouse is modeled as a randomly generated geometric graph.
For the entire collection see [Zbl 1390.68022].