Improved approximation algorithm for the Dense-3-Subhypergraph Problem

The study of Dense-3-Subhypergraph problem was initiated in Chlamtác et al. [7]. The input is a universe U and collection S of subsets of U , each of size 3, and a number k. The goal is to choose a set W of k elements from the universe, and maximize the number of sets, S ∈ S so that S ⊆W . The members in U are called vertices and the sets of S are called the hyperedges. This is the simplest extension into hyperedges of the case of sets of size 2 which is the well known Dense k-subgraph problem (see Kortsarz-Peleg [18]). The best known ratio for the Dense-3-Subhypergraph is O(n) by Chlamtác et al. [7]. We improve this ratio to n. More importantly, we contribute a new technique that gives a ratio of Õ(n/k) for the Dense-3-Subhypergraph improving the ratio of O(n/k) of Chlamtác et al. [7]. We prove that under the log density conjecture (see Bhaskara et al. [5]) the ratio cannot be better than Ω( √ n) and demonstrate some cases in which this optimum can be attained.