  We study a recently introduced generalization of the Vertex Cover (VC)
problem, called Power Vertex Cover (PVC). In this problem, each edge of the
input graph is supplied with a positive integer demand. A solution is an
assignment of (power) values to the vertices, so that for each edge one of its
endpoints has value as high as the demand, and the total sum of power values
assigned is minimized. We investigate how this generalization affects the
parameterized complexity of Vertex Cover. On the positive side, when
parameterized by the value of the optimal P, we give an O*(1.274^P)-time
branching algorithm (O* is used to hide factors polynomial in the input size),
and also an O*(1.325^P)-time algorithm for the more general asymmetric case of
the problem, where the demand of each edge may differ for its two endpoints.
When the parameter is the number of vertices k that receive positive value, we
give O*(1.619^k) and O*(k^k)-time algorithms for the symmetric and asymmetric
cases respectively, as well as a simple quadratic kernel for the asymmetric
case. We also show that PVC becomes significantly harder than classical VC when
parameterized by the graph's treewidth t. More specifically, we prove that
unless the ETH is false, there is no n^o(t)-time algorithm for PVC. We give a
method to overcome this hardness by designing an FPT approximation scheme which
gives a (1+epsilon)-approximation to the optimal solution in time FPT in
parameters t and 1/epsilon.
