Parameterized Power Vertex Cover

Abstract

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 complexity of Vertex Cover from the point of view of parameterized algorithms. On the positive side, when parameterized by the value of the optimal P, we give an $O^*(1.274^P)$ branching algorithm ($O^*$ is used to hide factors polynomial in the input size), and also an $O^*(1.325^P)$ 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 $O^*(k^k)$ 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 $n^{o(t)}$ algorithm for PVC. We give a method to overcome this hardness by designing an FPT approximation scheme which obtains a $(1+\epsilon )$-approximation to the optimal solution in time FPT in parameters t and $1/\epsilon$.