Metric Facility Location under Batched Random Order

Abstract

We study online metric facility location in the batched random-order model. An adversary partitions $n$ clients into $B$ nonempty batches, which arrive in uniformly random order. Each batch is revealed in full, and its clients must be connected to open facilities before the next batch arrives. Openings and assignments are irrevocable, and the objective is to minimize the total opening and connection cost. For both uniform and nonuniform opening costs, we prove that the tight competitive ratio is \[ Θ\!\left(\min\left\{\log(B+1), \sqrt{\log(n-B+2)}\right\}\right). \] Thus, the ratio is constant both when there is a single batch and when every batch is a singleton, but intermediate numbers of batches can require a $Θ(\sqrt{\log n})$ ratio. Our algorithms build on Meyerson's algorithm (2001) and do not need to know $n$ nor $B$. A key algorithmic novelty is that we process the clients in a batch one by one in uniform random order (hence neglecting the global properties of the batch). We present a novel analysis of this algorithm that hinges on the random order of the batches, as well as of the clients within each batch. We also provide a matching competitive ratio lower bound already for uniform costs, and even when the entire metric is known in advance.