Subexponential fixed-parameter algorithms for partial vector domination

Given a graph G=(V,E) of order n and an n-dimensional non-negative vector d=(d(1),d(2),…,d(n)), called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum S⊆V such that every vertex v in V∖S (resp., in V) has at least d(v) neighbors in S. The (total) vector domination is a generalization of many dominating set type problems, e.g., the (total) dominating set problem, the (total) k-dominating set problem (this k is different from the solution size), and so on, and subexponential fixed-parameter algorithms with respect to solution size for apex-minor-free graphs (so for planar graphs) are known. In this paper, we consider maximization versions of the problems; that is, for a given integer k, the goal is to find an S⊆V with size k that maximizes the total sum of satisfied demands. For these problems, we design subexponential fixed-parameter algorithms with respect to k for apex-minor-free graphs.