Vertex Cover on 4-Regular Hyper-graphs Is Hard to Approximate within 2 - ε

We consider the generalization of Minimum Vertex Cover to k-regular hyper-graphs, or equivalently, Minimum Hitting Set where all sets have size exactly k. We call this problem Minimum Ek Hitting Set. There is an easy k-approximation algorithm for this problem, and the best known algorithm is not much better; it approximates Minimum Ek Hitting Set within k − o(1) [4]. Turning to inapproximability results, for Minimum E2 Hitting Set, i.e, Minimum Vertex Cover, it was recently proven by Dinur and Safra [2] that Minimum Vertex Cover is NP-hard to approximate within 10 √ 5−21− ≈ 1.3606, improving on a previous result by Håstad [5] Since the general Minimum Hitting Set problem is equivalent to Minimum Set Cover, it follows by a result of Feige [3] that Minimum Hitting Set is “almost” NP-hard to approximate within a factor (1 − ) lnn for any > 0. This result is essentially tight since there is a well-known (1 + lnn)-approximation algorithm. The above mentioned inapproximability result for the general problem leads us to expect Minimum Ek Hitting Set to get harder to approximate as k grows. Indeed, recently Trevisan [7] proved that asymtoptically Minimum Ek Hitting Set is NP-hard to approximate within Ω(k). In this work we study the case k = 4, and we prove