Lecture 9 : Hardness of Max - E k - Indep . - Set and A . - k - Center February 12 , 2008

The proof of Theorem 1.1 uses a reduction from the Label-Cover(K,L) problem. Given an instance G = (U, V,E) of the Label-Cover(K,L) problem, we construct a k-regular hypergraphH. For each vertex v ∈ V , we add a corresponding “block” {0, 1}v of 2 vertices in H. We assign a p-biased weight to each vertex, with p = 1− 2 k − δ. For each pair of edges (u, v), (u, v′) ∈ E, we add a hyperedge on the set of vertices {A1, . . . , Ak/2, B ′ 1 , . . . , B v′ k/2} if and only if πv→u( ⋂k/2 i=1 A v i ) and πv′→u( ⋂k/2 i=1 B v′ i ) are disjoint sets of keys. For more details on the reduction and the intuition behind it, see Lecture 8. The reduction takes polynomial time. In the previous lecture, we used the reduction to prove the completeness of Theorem 1.1.