Subexponential Fixed-Parameter Algorithms for Partial Vector Domination

Abstract

Given a graph $G=(V,E)$ of order $n$ and an $n$-dimensional non-negative vector $\mathbf{d}=(d(1),d(2),\ldots ,d(n))$, called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum $S\subseteq V$ such that every vertex $v$ in $V\setminus S$ (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 dominating set problem, the $k$-tuple 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 $S\subseteq V$ 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.

This work is partially supported by KAKENHI No. 23500022, 24700001, 24106004, 25104521 and 25106508, the Kayamori Foundation of Informational Science Advancement and The Asahi Glass Foundation.