Hitting Sets Online and Vertex Ranking

Abstract

We consider the problem of hitting sets online for several families of hypergraphs.

Given a graph G = (V,E), let H = (V,R) denote the hypergraph whose hyperedges are the 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 online algorithm for H, we can use it as a black box in order to compute a vertex ranking for G with at most k colors. 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 and more general bound of Alon et al.

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. We obtain an O(logn)-competitive ratio. We also prove an Ω(logn) lower bound for the competitive ratio in this setting. The second hypergraph is defined by subsets of a given set of n points in the plane induced by unit discs. Since the number of subsets in this setting is O(n ²), the competitive ratio obtained by Alon et al. is O(log² n). We introduce an algorithm with O(logn)-competitive ratio. We also show that any online algorithm for this problem has a competitive ratio of Ω(logn), and hence our algorithm is optimal.