  Optimizing parameters of Two-Prover-One-Round Game (2P1R) is an important
task in PCPs literature as it would imply a smaller PCP with the same or
stronger soundness. While this is a basic question in PCPs community, the
connection between the parameters of PCPs and hardness of approximations is
sometime obscure to approximation algorithm community. In this paper, we
investigate the connection between the parameters of 2P1R and the hardness of
approximating the class of so-called connectivity problems, which includes as
subclasses the survivable network design and (multi)cut problems. Based on
recent development on 2P1R by Chan (ECCC 2011) and several techniques in PCPs
literature, we improve hardness results of some connectivity problems that are
in the form $k^\sigma$, for some (very) small constant $\sigma>0$, to hardness
results of the form $k^c$ for some explicit constant $c$, where $k$ is a
connectivity parameter. In addition, we show how to convert these hardness into
hardness results of the form $D^{c'}$, where $D$ is the number of demand pairs
(or the number of terminals).
  Thus, we give improved hardness results of k^{1/2-\epsilon} and
k^{1/10-\epsilon} for the root $k$-connectivity problem on directed and
undirected graphs, k^{1/6-\epsilon} for the vertex-connectivity survivable
network design problem on undirected graphs, and k^{1/6-\epsilon} for the
vertex-connectivity $k$-route cut problem on undirected graphs.
