Hitting sets online and unique-max coloring

We consider the problem of hitting sets online. The hypergraph is known in advance, and the hyperedges to be stabbed are input one-by-one in an online fashion. The online algorithm must stab each hyperedge upon arrival. The best known competitive ratio for hitting sets online by Alon et al. (2009) is O(lognlogm) for general hypergraphs, where n and m denote the number of vertices and the number of hyperedges, respectively. In this paper we provide the following results: 1. We consider hypergraphs in which the union of two intersecting hyperedges is also a hyperedge. Our main result for such hypergraphs is as follows: We consider a recently studied hypergraph coloring notion referred to as "unique-maximum coloring" and show that the competitive ratio of the online hitting set problem is at most the unique-maximum chromatic number and at least this number minus one. 2. Given a graph G=(V,E), let H=(V,R) denote the hypergraph whose hyperedges are subsets U⊆V such that the induced subgraph G[U] is connected. We establish a new connection between the best competitive ratio for the online hitting set problem in H and a well studied graph coloring notion referred to as the "vertex ranking number" of G. This connection states that these two parameters are equal. Moreover, this equivalence is constructive. As a corollary, we obtain optimal online hitting set algorithms for many such hypergraphs including those realized by planar graphs, graphs with bounded tree width, trees, etc. This improves the best previously known general bound of Alon et al. (2009). 3. We also consider two geometrically defined hypergraphs. The first one is defined by subsets of a given set of n points in the Euclidean plane that are induced by half-planes. The second hypergraph is defined by subsets of a given set of n points in the plane induced by unit discs. For these hypergraphs, the competitive ratio obtained by Alon et al. is O(log²n). For both cases, we introduce an algorithm with O(logn)-competitive ratio. We also show that any online algorithm for both settings has a competitive ratio of Ω(logn), and hence our algorithms are optimal.