Hitting Sets Online and Unique-Max Coloring

We consider the problem of hitting sets online. The hypergraph (i.e., rangespace consisting of points and ranges) is known in advance, and the ranges to be stabbed are input one-by-one in an online fashion. The online algorithm must stab each range upon arrival. An online algorithm may add points to the hitting set but may not remove already chosen points. The goal is to use the smallest number of points. The best known competitive ratio for hitting sets online by Alon et al. [AAA+09] is O(log n · logm) for general hypergraphs, where n and m denote the number of points and the number of ranges, respectively. We consider hypergraphs in which the union of two intersecting ranges is also a range. Our main result for such hypergraphs is as follows. The competitive ratio of the online hitting set problem is at most the unique-max number and at least this number minus one. 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 the vertex ranking number of G. This connection states that these two parameters are equal. Moreover, this equivalence is algorithmic in the sense, that given an algorithm to compute a vertex ranking of G with k colors, one can use this algorithm as a black-box in order to design a k-competitive deterministic online hitting set algorithm for H . Also, given a deterministic k-competitive School of Electrical Engineering, Tel-Aviv Univ., Tel-Aviv 69978, Israel. Mathematics Department, Ben-Gurion University of the Negev, Be’er Sheva 84105, Israel.