Lower bound for subset k connectivity for a set

1 The MinRep problem The challenge of making k << |T | is highly non trivial. The difficulty is among other things the need for reduction for subset k-connectivity. The construction is essentially new and has only minor intersection with the one from [KKL]. Generally speaking, we must make the reduction local. For example in the sense that we mainly connect vertices at distance 2. We describe the reduction by an ’almost’ self contained graph problem called MinRep [Kor01]. There are 5 parameters, m,n, l, qA, qB that help define the MinRep instance. The values of these parameters are listed below and follow from the most standard one round two provers protocol. See for example [CCK]. The input consists of a bipartite graph G(A,B,E), with an explicit partitioning of each of A and B into A = ⋃qA i=1 Ai, |Ai| = 7 l for all i and B = ⋃qB j=1 Bj, |Bj| = 2 , for every j. The bipartite graph G induces a supergraph H as follows. The supervertices (i.e., the vertices of H) are the qA+ qB sets Ai and Bj. A superedge (an Department of Computer Sciences, Rutgers University, Camden, NJ 08102, USA. Supported in part by NSF grant number 0829959 Email: [email protected] Department of Computer Science. The Open University, Raanana, Israel Email: [email protected]